Particle Adhesion Fundamentals and Bulk Powder Consolidation +

The fundamentals of particulate solids consolidation and flow behaviour using a reasonable combination of particle and continuum mechanics are explained. By means of the model "stiff particles with soft contacts" the combined influence of elastic, plastic and viscoplastic repulsion in particle contacts is derived. Consequently, contact normal force displacement FN(hK) and adhesion force models FH(FN) are presented to describe the stationary, instantaneous and time consolidation behaviour in particle contacts as well as in the bulk. On this particle mechanical basis, the stationary, instantaneous and time yield loci as well as uniaxial compressive strength O"c ( 0"1), effective angle of internal friction tf>e( 0"1), bulk density pb( 0"1) are derived and shown for a very cohesive sub-micron titania powder. Finally, these models in combination with accurate shear cell test results are used as constitutive functions for computer aided silo design for reliable flow and pressure calculations.


Introduction
The well-known flow problems of cohesive particulate solids in storage and transportation containers, conveyors or process apparatuses leads to bridging, channelling, oscillating mass flow rates and particle characteristics with feeding and dosing problems as well as undesired effects like widely spread residence time distribution, fertiliser time consolidation process or caking, chemical conversions, food deterioration and finally insufficient element and system reliability of solid processing plants.Taking into account this list of selected technical problems and hazards, it is really necessary to deal with the fundamentals of particulate solids consolidation and flow behaviour, namely from using a reasonable combination of particle and continuum mechanics.

Fig. 1
Particle contact approaching, elastic, elastic-plastic deformation and removing First of all, Rumpf et al. (25) have developed a constitutive model approach for describing the linear increasing of adhesion force F H, mainly for plastic contact deformation: With another prerequisite and derivation Molerus (12) obtained an equivalent formula: The adhesion force FHo without additional consolidation (FN=O) can be approached as a single spheresphere-contact, see Fig. la).This soft particle contact is flattening by the external normal force FN to a plateplate-contact, Fig. lc).The coefficient Kp describes a dimensionless ratio of attractive Van Der Waals pressure Pvdw for a plate-plate model to repulsive particle micro-hardness p 1 (resistance against plastic deformation) 158 and is to be called here as a plastic repulsion coefficient.The characteristic adhesion separation distance lies in a molecular scale (atomic centre to centre distance) and amounts to about a=aF=o=0.3-0.4 nm.It depends mainly on the properties of liquid-equivalent packed adsorbed layers and can be estimated for a molecular force equilibrium or interaction potential minimum -dU/da=F=O=Fattraction+Frepuision• The Hamaker constant solid-liquid-solid CH,sls ace. to Lifschitz theory is related to continuous media, dependent on their permittivities (dielectric constants) and refractive indices, see Israelachvili (29).
If the contact radius r is small compared with the particle size d, an elliptic pressure distribution Pe 1 (r) is found, see Hertz (15).Schubert (23) has combined the elastic and plastic contact strain expressed with the annular elastic Ae 1 and circular plastic Ap 1 contact area: Taking into account these Eqs.(l), ( 2), (4) as well as the model from Thornton (28) and superposition pro-vided, the particle contact force equilibrium between attraction (-) and elastic as we11 as plastic repulsion or force response ( +) delivers a very useful linear force displacement model for KA =con st. with the particle centre approach of both partners hK (32), Fig. 2: (6) Thus, the contact stiffness decreases with sma11er size especia11y for cohesive fine powders and nanoparticles, predominant plastic yielding behaviour provided (32): (7) 25 But if we consider Eq.( 9) a slightly nonlinear (progressively increasing) curve is obtained.
The dominant linear elastic-plastic deformation range between rea11y tested yield loci levels YLl to YL 4 is to be demonstrated in Fig. 2. The origin of this diagram hK=O is equivalent to the characteristic adhesion separation for a direct contact (atomic centre to centre distance) and can be estimated for a molecular force equilibrium a=ao=aF=O• Provided that these molecular contacts are stiff enough compared with the soft mesoscopic particle contact behaviour influenced by mobile adsorption layers due to molecular rearrangement, this separation aF~o is assumed to be constant during loading and unloading in the interesting macroscopic pressure range of cr=O.l-100kPa.After loading 0-Y the contact is elastica1ly compacted with an approximated circular contact area, see   with plastic yielding, see Fig. 1c).Now, the combined elastic-plastic yield boundary of the plate-plate contact is achieved, see Eq.( 6).This displacement is to be expressed with the annular elastic Ae 1 (thickness rK.ei) and circular plastic Apl (radius rK.pi) contact area, see Fig. 1c).For comparison, a dashed line for the flowability limit ffc=2 (very cohesive ace. to Jenike (2)) is plotted additionally (32), see Fig. 2.After unloading U-E the contact recovers elastically in the compression mode and remains with an perfect plastic displacement hK.E• Below point E left the tension mode begins.Between U-E-A the contact recovers probably elastically along a supplemented Hertzian parabolic curvature up to a displacement hK.A• Along A-U the contact could be reloaded.If we apply a certain pull-off force F N.z=-F H. A. here negative, the adhesion (failure) boundary at point A is reached and the contact plates are failing and removing with the increasing distance a=aF=o+ hK.A-hK.This actual particle separation is to be considered for the calculation by means of a hyperbolic adhesion force curve FN.z=-FH.A oca-3 with the plate-plate model, see Eq.( 3).These new models in Fig. 2 follow Molerus' (11), Schubert's (23), Maugis' ( 26) and especially Thornton's (28) example.The slopes of plastic curves are a measure of irreversible particle contact stiffness or softness, resp.Thus the dimensionless contact consolidation coefficient (strain characteristic) K is read here as the slope of adhesion force, Fig. 3, influenced by predominant plastic contact failure.(8) The elastic-plastic contact area coefficient KA represents the ratio of plastic particle contact deformation area Ap 1 to total contact deformation area AK=Ap 1 + Ae 1 with the centre approach hK.t for incipient yielding at Pel (r= 0) = Pmax =PI: The pure elastic contact deformation Ap 1 =0, KA=2/3, has no relevance for fine cohesive particles and should be excluded here.Commonly, for pure plastic contact deformation Ae 1 =0 or AK=Ap!.KA=1 is obtained.From Eq.( 5), a linear model for the adhesion force FH as function of normal force F N is obtained again (32): The contact consolidation coefficient K is a measure of irreversible particle contact stiffness or softness as well, see Fig. 3.A small slope stands for low adhesion level FH""FHo because of stiff particle contacts, but a large inclination means soft contacts or consequently, a cohesive powder flow behaviour, see Fig. 6 as well.This model considers additionally the flattening of soft particle contacts caused by acting an adhesion force K • F HO• Herewith, the adhesion force F Ho for Ft-;=0 considers an essential characteristic micro roughness heighthr<d particle size (Schubert (30)): The intersection of function (11) with abscissa (FH=O) in the negative extension range of consolidation force F N. Fig. 3, is surprisingly independent of the Hamaker constant CH,sls: Considering the model prerequisites for cohesive powders, this minimum normal (tensile) force limit FN,z combines the influences of a particle contact hardness or micro-yield strength p 1 =3•cr 1 (cr 1 yield strength in tension) and particle separation distance distribution characterised by a particle roughness height hr as well as molecular centre separation distance aF=O• Obviously, this value characterises the contact softness with respect to a small asperity height hr as well, compare Eq. (7).It corresponds to an abscissa intersection O\z of the constitutive consolidation function crc(cr 1 ), see Fig. 6 and Fig. 8 below.
A term for deformation rate influence on adhesion force in a particle contact is to be inserted (14) with a viscoplastic contact consolidation coefficient K 1 =attraction/repulsion force ratio as a dimensionless combination of attractive contact strength cra = Pvdw and repulsive particle contact viscosity llv/t = p 1 , i.e. viscoplastic stiffness, analogous to plastic deformation, Fig. 3.
For example, the driving potential in the main case of viscoplastic contact deformation or particle fusion resp., is given by means of free surface energy <J 55 • On the one hand, the influence of the pre-consolidation normal force F : --; on the contact circle radius ratio rvis/d was derived from Rumpf et al. ( 25): The adhesion force F Ht is set proportional to the tensile strength of flowing material.This is created either by means of liquid-equivalent adsorption layer bridges with Van Der Waals and hydrogen bondings or by Van Der Waals solid bridges <Jzs = Ga and lls = r)v, with small melting point (0.75-0.9) •Tm for particle fusion in the contact zone: (17) Generally, this corresponds to linear particle contact constitutive model, Eq.( 14), with the viscoplastic repulsion or contact consolidation coefficient K 1 as a slope of adhesion force when a normal force F : --; acts in the deformed contact (33,34,35,36): The particle viscosity lls decreases with temperature increasing.This is to be described with a typical thermal kinetic expression for particle fusion with Jar mass, chain length and number of cross-links for polymers (31).
The median adhesion forces F~o and FHo.see Eq.( 12), of a direct spherical contact correspond each other if the so-called Derjaguin approximation ( 29) is valid (aF=o <<d) 1 CH,svs 24•rc•a}=o (22) and the particle bondings are comparatively weak, e.g.Van Der Waals interaction (CH.svsHamaker constant solid-vacuum-solid).But, these bonds are strong enough to disturb essentially reliable flow of particulate solids.
In opposition to time invariable plastic contact deformation, all parameters depend on time.Therefore superposition provided, the total adhesion force F Htot consists of a instantaneous FH and a time influenced component FHt, Fig. 3: (23) In this context, the total adhesion force F Htot is a function of adhesion force F Ho without any deformation or pre-consolidation in a very closed sphere-sphere contact plus a consolidation term (K+KJ • (F:--;+ FHo) with a normal force in the flattened plate-plate particle contact.Finally, this Eq.( 23) can be interpreted as a general linear particle contact constitutive model, i.e. linear in forces and stresses, but non-linear concerning material characteristics, Table 1.Generally, this adhesion force level, see Fig. 3, amounts up to 10 5 -10 6 fold of particle weight for very cohesive fine particles.Additionally, for moist powders, the liquid bridge bonding forces are strength determined (33,38,39).

Particle contact Failure and cohesive Powder Flow criteria
Regarding the formulation of failure conditions at the particle contacts, we can obviously follow the Molerus theory ( 14), but here with a general supplement for the particle contact constitutive models Eqs.(11), ( 14) and (23), see in detail (32).The inclina-tion of radius and centre contact force components FR, FM and normal vectors are described with an angle a, see contact failure conditions in Table 1.It should be noted that the stressing pre-history of a cohesive powder flow is stationary (steady-state) and delivers significantly a cohesive stationary yield locus in radius-centre-stress coordinates, (24) with the isostatic tensile strength 0 0 obtained from the adhesion force FHo, see Eq.( 12).
From it, the stress dependent effective angle of internal friction <!le ace. to Jenike as a slope of cohesionless effective yield locus follows obviously, see Fig. 4: If the major principal stress 0 1 reaches the stationary uniaxial compressive strength 0c.st.-------------------1 stationary angle of internal friction <p" major principal stress cr 1 during consolidation normal stress cr Stationary yield locus (33) and effective yield locus ace. to ] enike (2) For the relation between the angle of internal friction <pi (slope of instantaneous yield locus) and the stationary angle of internal friction <!lst following relation is used (12): The softer the particle contacts, the larger the difference between these friction angles are and consequently the more cohesive the powder behave.Therefore considering Eq.( 23), the new relation between the time dependent angle of internal friction <!lit (slope of time yield locus) and the time invariable stationary angle of internal friction <!lst (slope of stationary yield locus) is defined as Fig. 5: Now, the angle of internal friction of a time consolidation <flit is to be expressed ( 33): (30) 1 2 crzs tan <fli First of it, with this Eq.(30) following predictions are possible (33,34,35,36,37), Fig. 5: (1) If no time consolidation occurs t=O, both friction angles are equivalent <flit=<fli• The non-linear and linear instantaneous yield loci in radius-centrecoordinates are obtained: .
[ l(tan<pst 0M,st-0M ) ( cr, normal stress cr=F~/ A Fig. 5 Characteristics of instantaneous, stationary and time yield locus (3) For t~ oo follows <rit~o, that means, the time yield locus is a parallel line to the cr-axis, i.e. failure criterion of ideal plasticity by Tresca.The bulk material is hardening to a complete solid state with plastic failure conditions as a limitation.The non-linear and linear yield loci in T-cr-coordinates are shown in Table 1 and Fig. 5.These instantaneous yield loci, complementary consolidation loci and the stationary yield locus are completely described only with three material parameters plus the characteristic pre-consolidation (average pressure) influence (32): (1) <pi-incipient particle friction of failing contacts, i.e.Coulomb friction; (2) <rst -steady-state particle friction of failing contacts, increasing adhesion by means of contact flattening expressed with the contact consolidation coefficient K, or by friction angles sin(<p 8 t-<pi) and (sin<pst-sin<pi) see Eqs.( 31), ( 32) and (36).The softer the particle contacts, the larger the difference between these friction angles are and consequently the more cohesive the powder behave; (3) cr 0 -isostatic tensile strength of unconsolidated particle contacts, characteristic cohesion force in an unconsolidated powder, see Eq.( 37); (4) crM,st-previous consolidation influence of an additional normal force in a particle contact, characteristic centre stress of Mohr circle of pre-consolidation state directly related to powder bulk density.This average pressure influences the increasing isostatic tensile strength of yield loci by means of cohesive steady-state flow as the stressing pre-history of a powder.With the derivation of time yield locus the uniaxial compressive strength (unconfined yield strength) cret is found as function of the major principal stress cr 1 comparable with a linear bulk material constitutive model, Table 1: The slope au and the intersection of cre-axis cret,o are time dependent, Eq.( 30).The abscissa intersection cr1,2 of linear consolidation constitutive function cre(cr 1 ), corresponds to the FN,z value of contact consolidation function ace. to Eq.( 13) and Fig. 3.
Again, the following predictions are to be advanced,: -sin2 (<p,, -<p;) -tan <p; •sin (<p"-<p;)] -sin 2 (<p" -<p,) -tan <p ,, •sin (<p" -<p ;,)] (2) But if t>O the angle of internal friction during time consolidation decreases <l>it<<t>i and the slope au increases.(3) Fort~ x is <l>it~o.that means, the slope follows au~ 1.This is the largest slope considering the model prerequisites of an only viscoplastic flow.If the first derivative is greater then one au=dO'ctl da 1 > 1 a non-linear relation of particle contact deformation and consequently other physicochemical effects of irreversible bulk material consolidation should be considered.(4) Notice that for t~x the intersection of O'ct-axis O'ct.o achieves a upper limit, which is only dependent on surface en~rgy 0 55 and particle size and not from time and viscosity:

Powder Flowability and Compressibility
Assessing the flow behaviour of a powder, Eq.( 35) shows that the flow function ace. to Jenike (2) is not constant and depends on the consolidation stress level cr 1 : O"ct 2 sin <jlst -sin <!'it+sin <!' st • (l+sin <!'it) •cro/ 0"1 But roughly we can write for a smaii intersection with the ordinate Gc.o, i.e. isostatic tensile strength cr 0 ~0 near zero, the stationary angle of internal friction is equivalent to the effective angle <pst "-"' <!' e and J enike's (2) formula is obtained in order to demonstrate the general model validity: Thus, the semi-empirical classification by means of the flow function introduced by Jenike ( 2) is adopted here with a certain particle mechanical sense completion, Table 2: The class "non flowing" is characterised by the fact that the unconfined yield strength O"ct is higher than the consolidation stress cr 1 and thus in case of time consolidation, caking, cementation or hardening the powder has been agglomerated to solid state (38).
Obviously, the flow behaviour is mainly influenced by the difference between the friction angles, Eq.( 40), as a measure for the adhesion force slope K in the general linear particle contact constitutive model, Eq.( 23).Therefore we can recalculate these coefficients from flow function measurements: A characteristic value K=0.77 for <pi=30o of a very cohesive powder is included in the force displacement, Fig. 2, as well as adhesion force diagram, Fig. 3, and shows directly the correlation between strength and force enhancement with pre-consolidation, Table 2. Due to the consolidation function in Fig. 6, a small slope stands for a free flowing particulate solid with very low adhesion level because of stiff particle contacts, but a large inclination means a very cohesive powder flow behaviour because of soft particle contacts, Fig. 3.
Obviously, the finer the particles the "softer" the contacts and the more cohesive the powder (33,42).Kohler (43) has experimentaily confirmed this thesis for alumina powders (a-A]z0 3 ) down to the submicron range (crc,o""'Const.=2kPa, d 50 median particle size in 11m): (42) Analogously to adiabatic gas law p•Vl(ad=const., a differential equation for isentropic compressibility of a powder dS=O, i.e. remaining stochastic homogeneous packing without a regular order in the continuum, is to be derived: The total pressure including particle interaction p= 0":.1,st+cr 0 should be equivalent to a pressure term with molecular interaction CVm molar volume) (44) in Van Der Waals equation of state to be valid near gas condensation point.A loose powder packing is obtained Pb=Pb.o, if only particles are interacting without an external consolidation stress cr:.f.st=O, e.g.particle weight compensation by a fluid drag, and Eq.( 43) is solved: Therefore, this physicaily based compressibility index n = 1/Kad lies between n=O, i.e. incompressible stiff bulk material and n = 1, i.e. ideal (gas) compressibility index, see Fig. 7 above.
For hopper design purposes in powder mechanics the major principle stress cr 1 during preconsolidation is used instead of the centre stress crM.st•Hence we replace the total pressure p=cr 1 +crl.Z with the new abscissa intersection O"tz in the negative tensile or puii-off range of the consolidation function crc=f(cr 1 ), Eq.(36) and Fig. 7, and obtain this function: Considering the predominant plastic and viscoplastic particle contact deformation and rearrangement in the stochastic homogeneous packing of a cohesive powder, following values of compressibility index are to be suggested, Table 3: This approach should express the enormous problems concerning reliable flow of powders which are tending to time consolidation, hardening and caking.Consequently, discharging aids should be applied in handling practice (33).
The essential consolidation functions necessary for reliable design are collected in Fig. 8.For the cohesive steady-state flow cr 1 =crc,st.Fig. 4 and Eq. ( 27), the flow factor is ff= 1 and a minimum outlet width bmin.st<bmin(instantaneous flow) is obtained which prevents bridging during the stationary hopper operation.
Both, shear test results -accurate measurements provided -evaluated with these combined particle and continuum mechanical approach, are used as constitutive functions for computer aided silo design for reliable flow (37) on the one hand.On the other hand a supplemented slice-element standard method (41,42) is used for pressure calculations.Considering the reliable physical basis of the Pb(cr 1 ) and <pe(cr 1 ) functions for example, these can be suitably extrapolated using pressure calculations for large silos with more than 1000 m 3 storage capacity (44).

Conclusions
Taking into consideration all the different properties of cohesive to very cohesive powders tested (particle size distribution, moisture content, material properties etc.), the model fit can be characterised as satisfactory to good.Thus, the model has proved its effectiveness and can be accordingly applied in reliable silo design for flow and pressure calculation.
Obviously, recommendations are to be elaborated with respect to the powder product design for processing, logistics, transportation, distribution and consumption, see Borho et al. ( 45) as well.

Acknowledgements
The author would like to acknowledge his co-workers Dr. S. Arnan, Dr. T. Groger, S. Schubert, B. Reichmann and Th.Kollmann for their experimental contributions, relevant information and theoretical tips.

1 Fig. 8
Fig. 8Consolidation functions of a cohesive powder for reliable design

Symbols a : separation a 1 :
slope of crc(cr 1 ) consolidation function A : area, particle contact area b : outlet width CH : Hamaker constant d : particle size KONA No.18 (2000) factor ace. to J enike ffc : flow function ace. to J enike g : gravity acceleration m : mass, hopper shape factor p radius a : failure angle ~ : auxiliary failure angle function y : shear deformation rate gradient :porosity £ £ : deformation rate gradient 11 : viscosity TJv : viscoplastic yield strength of particle contact K : contact consolidation coefficient e : hopper angle <p : angle of friction p : density a : normal stress O"a : adhesion strength cr 1 : major principal stress a 0 : isostatic tensile strength

Table 1
(37)d characteristics from Particle Mechanics point of view(37)

Table 2
Flowability assessment and elastic-plastic contact consoli-