mechanics of nanoparticle adhesion — a continuum approach¤ge/mitps817-p-1090.pdf · hazards, it...

Particles on Surfaces 8: Detection, Adhesion and Removal, pp. 1–47 Ed. K.L. Mittal © VSP 2003

Mechanics of nanoparticle adhesion — A continuum approach

JÜRGEN TOMAS∗ Mechanical Process Engineering, Department of Process Engineering and Systems Engineering, Otto-von-Guericke-University, Universitätsplatz 2, D-39106 Magdeburg, Germany

Abstract—The fundamentals of particle-particle adhesion are presented using continuum mechan-ics approaches. The models for elastic (Hertz, Huber, Cattaneo, Mindlin and Deresiewicz), elastic-adhesion (Derjagin, Bradley, Johnson), plastic-adhesion (Krupp, Molerus, Johnson, Maugis and Pol-lock) contact deformation response of a single, normal or tangential loaded, isotropic, smooth con-tact of two spheres are discussed. The force-displacement behaviors of elastic–plastic (Schubert, Thornton), elastic–dissipative (Sadd), plastic–dissipative (Walton) and viscoplastic–adhesion (Rumpf) contacts are also shown. Based on these theories, a general approach for the time and de-formation rate dependent and combined viscoelastic, plastic, viscoplastic, adhesion and dissipative behaviors of a spherical particle contact is derived and explained. The decreasing contact stiffness with decreasing particle diameter is the major reason for adhesion effects at nanoscale. Using the model “stiff particles with soft contacts”, the combined influence of elastic-plastic and viscoplastic repulsions in a characteristic (averaged) particle contact is shown. The attractive particle adhesion term is described by a sphere–sphere model for van der Waals forces without any contact deforma-tion and a plate–plate model for this micro-contact flattening is presented. Various contact deforma-tion paths for loading, unloading, reloading and contact detachment are discussed. Thus, the varying adhesion forces between particles depend directly on this “frozen” irreversible deformation, the so-called contact pre-consolidation history. Finally, for colliding particles the correlation between par-ticle impact velocity and contact deformation response is obtained using energy balance. This con-stitutive model approach is generally applicable for solid micro- or nanocontacts but has been shown here for dry titania nanoparticles.

Keywords: Powder; particle mechanics; contact behavior; constitutive models; adhesion force; nanoparticles.

1. INTRODUCTION

In terms of particle processing and product handling, the well-known flow prob-lems of cohesive powders in storage and transportation containers, conveyors or process apparatuses include bridging, channeling and oscillating mass flow rates. In addition, flow problems are related to particle characteristics associated with

∗Phone: (49-391) 67-18783, Fax: (49-391) 67-11160, E-mail: [email protected]

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feeding and dosing, as well as undesired effects such as widely spread residence time distribution, time consolidation or caking, chemical conversions and deterio-ration of bioparticles. Finally, insufficient apparatus and system reliability of powder processing plants are also related to these flow problems. The rapid in-creasing production of cohesive to very cohesive nanopowders, e.g., very adher-ing pigment particles, micro-carriers in biotechnology or medicine, auxiliary ma-terials in catalysis, chromatography or silicon wafer polishing, make these problems much serious. Taking into account this list of technical problems and hazards, it is essential to deal with the fundamentals of particle adhesion, powder consolidation and flow, i.e. to develop a reasonable combination of particle and continuum mechanics.

The well-known failure hypotheses of Tresca, Coulomb and Mohr and Drucker and Prager (in Refs. [1, 2]), the yield locus concept of Jenike [3, 4] and Schwedes [5], the Warren–Spring equations [6–10], and the approach by Tüzün [12], etc., were supplemented by Molerus [13–16] to describe the cohesive, steady-state flow criterion. Nedderman [17, 18], Jenkins [19] and others discussed the rapid and collisional flow of non-adhering particles, as well as Tardos [20] discussed the frictional flow for compressible powders without any cohesion from the fluid mechanics point of view. Additionally, the simulation of particle dynamics of free flowing granular media is increasingly used, see, e.g., Cundall [21], Campbell [22], Walton [23, 24], Herrmann [25] and Thornton [26].

Additionally, particle adhesion effects are related to undesired powder blocking at conveyer transfer chutes or in pneumatic pipe bends [27] in powder handling and transportation, to desired particle cake formation on filter media [28, 29], to wear effects of adhering solid surfaces [30, 31], fouling in membrane filtration, fine particle deposition in lungs, formulation of particulate products [32–35] or to surface cleaning of silicon wafers [36–40, 133], etc.

The force–displacement behaviors of elastic, elastic–adhesion, plastic–adhesion, elastic–plastic, elastic–dissipative, plastic–dissipative and viscoplastic–adhesion contacts are shown. Based on these individual theories, a general ap-proach for the time and deformation rate dependent and combined viscoelastic, plastic, viscoplastic, adhesion and dissipative behaviors of a spherical particle contact is derived and explained.

2. PARTICLE CONTACT CONSTITUTIVE MODELS

In terms of particle technology, powder processing and handling, Molerus [13, 14] explained the consolidation and non-rapid flow of dry, fine and cohesive powders (particle diameter d < 10 µm) in terms of the adhesion forces at particle contacts. In principle, there are four essential mechanical deformation effects in particle–surface contacts and their force–response (stress-strain) behavior can be explained as follows (Table 1):

Mechanics of nanoparticle adhesion — A continuum approach 3

(1) elastic contact deformation (Hertz [41], Huber [42], Derjaguin [43], Bradley [44, 45], Cattaneo [46], Mindlin [47], Sperling [48], Krupp [49], Greenwood [50], Johnson [51], Dahneke [52], Thornton [53, 54], Sadd [55]), which is re-versible, independent of deformation rate and consolidation time effects and valid for all particulate solids;

(2) plastic contact deformation with adhesion (Derjaguin [43], Krupp [56], Schu-bert [57], Molerus [13, 14], Maugis [58], Walton [59] and Thornton [60]), which is irreversible, deformation rate and consolidation time independent, e.g. mineral powders;

(3) viscoelastic contact deformation (Yang [61], Krupp [49], Rumpf et al. [62] and Sadd [55]), which is reversible and dependent on deformation rate and consolidation time, e.g., bio-particles;

(4) viscoplastic contact deformation (Rumpf et al. [62]), which is irreversible and dependent on deformation rate and consolidation time, e.g., nanoparticles fu-sion.

This paper is intended to focus on a characteristic, soft contact of two isotropic, stiff, linear elastic, smooth, mono-disperse spherical particles. Thus, this soft or compliant contact displacement is assumed to be small (hK/d << 1) compared to the diameter of the stiff particle. The contact area consists of a representative number of molecules. Hence, continuum approaches are only used here to de-scribe the force-displacement behavior in terms of nanomechanics. The micro-scopic particle shape remains invariant during the dynamic stressing and contact deformation at this nanoscale. In powder processing, these particles are manufac-tured from uniform material in the bulk phase. These prerequisites are assumed to be suitable for the mechanics of dry nanoparticle contacts in many cases of indus-trial practice.

2.1. Elastic, plastic and viscoplastic contact deformations

2.1.1. Elastic contact displacement For a single elastic contact of two spheres 1 and 2 with a maximum contact circle radius rK,el but small compared with the particle diameter d1 or d2, an elliptic pres-sure distribution pel(rK) is assumed, Hertz [41]:

22

el K

max K el1

� �� = � �� ,

p r–

p r (1)

With the maximum pressure p(rK = 0) = pmax in the center of contact circle at depth z = 0,

Nmax 2

K el

3

2

⋅=

⋅ ⋅ ,

Fp

rπ (2)

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and the median particle radius r1,2 (characteristic radius of contact surface curva-ture) (Fig. 1),

1

1 21 2

1 1–

,rr r

� �= +� ��

(3)

and the average material stiffness (E, modulus of elasticity; ν, Poisson ratio)

12 21 2

1 2

1 12*E

E Eν ν

−� �− −

= ⋅ +� ��

(4)

one can calculate the correlation between normal force FN and maximum contact radius rK,el:

3 1 2 NK el

3

2,

, *

r Fr

E

⋅ ⋅=

⋅ (5)

Considering surface displacement out of the contact zone (for details, see Huber [42]) the so-called particle center approach or height of overlap of both particles hK is [41]:

2K K el 1 2, ,h r / r= (6)

Substitution of Eq. (6) in Eq. (5) results in a non-linear relation between elastic contact force and deformation [41] (Fig. 1):

3N 1 2 K

23

*,F E r h= ⋅ ⋅ ⋅ (7)

Eq. (7) is shown as the dashed curve marked Hertz. The maximum pressure

pmax, Eq. (2), is 1.5 times the average pressure 2N K el( )⋅ ,F / rπ on the contact area

and lies below the micro-yield strength pf. Because of surface bending and, con-sequently, the opportunity for unconfined yield at the surface perimeter outside of the contact circle rK ≥ rK,el (Fig. 1b), a maximum tensile stress is found [42]

t max max0 15≈ − ⋅, . pσ (here negative because of positive pressures in powder me-

chanics):

K K el

2t K

2max K el

1 23

,r r ,

rp r

σ ν≥

− ⋅= − ⋅ (8)

This critical stress for cracking of a brittle particle material with low tensile strength is smaller than the maximum shear stress max max0 31. pτ ≈ ⋅ according to

Eq. (11) which is found at the top of a virtual stressing cone below the contact

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Figure 1. Characteristic spherical particle contact deformation. (a) Approach and (b) elastic contact deformation (titania, primary particles d = 20–300 nm, surface diameter dS = 200 nm, median parti-cle diameter d50,3 = 610 nm, specific surface area AS,m = 12 m²/g, solid density ρs = 3870 kg/m3, sur-face moisture XW = 0.4%, temperature θ = 20°C) [148]. Pressure and compression are defined as positive but tension and extension are negative. The origin of this diagram (hK = 0) is equivalent to the characteristic adhesion separation for direct contact (atomic center to center distance), and can be estimated for a molecular force equilibrium a = a0 = aF=0. After approaching from an infinite dis-tance –∞ to this minimum separation aF=0 the sphere–sphere contact without any contact deforma-tion is formed by the attractive adhesion force FH0 (the so-called “jump in”). Then the contact may be loaded FH0 – Y and, as a response, is elastically deformed with an approximate circular contact area due to the curve marked with Hertz (panel b). The tensile contribution of principal stresses ac-cording to Huber [42] at the perimeter of contact circle is neglected for the elliptic pressure distribu-tion, drawn below panel b.


area on the principal axis 2 2 2K 0r x y= + = in the depth of z ≈ rK,el/2. Combining

the major principal stress distribution, Eq. (9), σ1 = σz(z) at contact radius rK = 0

2K el

2 2max K el

( ) ,z

,

rzp r z

σ=

+, (9)

and the minor principal stress, Eq. (10), σ2 = σy = σt(z) [42]

( )K K el

2K el K elt

2 2max K elK el

( ) 1 1 1 arctan2

,

, ,

,r r ,

r rz zp r zr z

σν

≤

� �= − ⋅ + + ⋅ − ⋅� �

+ � �, (10)

the maximum shear stress inside a particle contact rK ≤ rK,el is obtained using the

Tresca hypothesis for plastic failure ( )max 1 2 2τ σ σ= − [1]:

K K el

2K el K el

2 2max K elK el

( ) 3 1 1 arctan4 2

,

, ,

,r r ,

r rz zp r zr z

τ ν≤

� �+= ⋅ − ⋅ − ⋅� �+ � �

(11)

This internal shear stress distribution becomes more and more critical for duc-tile or soft solids with a small transition to yield point, and consequently, plastic contact deformation like nanoparticles with very low stiffness, see Section 2.3.

Due to the parabolic curvature FN(hK), the particle contact becomes stiffer with increasing diameter r1,2, contact radius rK or displacement hK (kN is the contact stiffness in normal direction):

NN 1 2 K K

K

dd

* *,

Fk E r h E r

h= = ⋅ ⋅ = ⋅ (12)

The influence of a tangential force in a normal loaded spherical contact was considered by Cattaneo [46] and Mindlin [47, 63]. About this and complementary theories as well as loading, unloading and reloading hysteresis effects, one can find a detailed discussion by Thornton [53]. He has expressed this tangential con-tact force as [53, 54]:

( )T 1 2 K i N4 1 tan*,F G r h Fψ δ ψ ϕ= ⋅ ⋅ ⋅ ⋅ ⋅ ∆ ± − ⋅ ⋅∆ (13)

Here ∆δ is the tangential contact displacement, ψ the loading parameter de-pendent on loading, unloading and reloading, ϕi the angle of internal friction,

( )2 1G E ν= + the shear modulus, and the averaged shear modulus is given as:

1

1 2

1 2

2 22

–* – –

GG G

ν ν� �= ⋅ +� ��

(14)

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Thus, with ψ = 1 the ratio of the initial tangential stiffness

TT K

d4

d*F

k G rδ= = ⋅ ⋅ (15)

to the initial normal stiffness according to Eq. (12) is:

( )T

N

2 12

kk

νν

⋅ −= − (16)

Hence this ratio ranges from unity, for ν = 0, to 2/3, for ν = 0.5 [63], which is different from the common linear elastic behavior of a cylindrical rod.

2.1.2. Elastic displacement of an adhesion contact The adhesion in the normal loaded contact of spheres with elastic displacement will be additionally shown. For fine and stiff particles, the Derjaguin, Muller and Toporov (DMT) model [43, 65, 66] predicts that half of the interaction force FH,DMT/2 occurs outside in the annular area which is located at the perimeter closed by the contact, Eq. (17). This is in contrast to the Johnson, Kendall and Roberts (JKR) model [67], which assumes that all the interactions occur within the contact radius of the particles. The median adhesion force FH,DMT (index H,DMT) of a direct spherical contact can be expressed in terms of the work of adhesion WA, conventional surface energy γA or surface tension σsls as

A A sls2 2W γ σ= ⋅ = ⋅ . The index sls means particle surface-adsorption layers (with

liquid equivalent mechanical behavior) – particle surface interaction. If only mo-lecular interactions with separations near the contact contribute to the adhesion force then the so-called Derjaguin approximation [43] is valid

H DMT sls 1 24, ,F rπ σ= ⋅ ⋅ ⋅ , (17)

which corresponds to Bradley’s formula [44]. This surface tension σsls equals half the energy needed to separate two flat surfaces from an equilibrium contact dis-tance aF=0 to infinity [75]:

0

H slssls VdW 2

0

1 ( ) d2 24

F

,

a F

Cp a a

aσ

π=

∞

=

= − ⋅ =⋅ ⋅� (18)

The adhesion force per unit planar surface area or attractive pressure pVdW which is used here to describe the van der Waals interactions at contact is equiva-lent to a theoretical bond strength and can simply calculated as [75] (e.g., pVdW ≈ 3–600 MPa):

H sls slsVdW 3

00

4

6,

FF

Cp

aa

σπ ==

⋅= =

⋅ ⋅ (19)


Using this and for a comparison of the adhesion or bond strength, a dimen-sionless ratio of adhesion displacement (extension at contact detachment), ex-pressed as height of “neck” hN,T around the contact zone, to minimum molecular center separation aF=0 can be defined as

1 32

N T 1 2 slsT 2 30

0

4/

, ,

*FF

h r

a E a

σΦ

==

� �⋅ ⋅� �= =� �⋅� �

, (20)

which was first introduced by Tabor [74] and later modified and discussed by Muller [66] and Maugis [76]. The DMT model works for very small and stiff par-ticles ΦT < 0.1 [66, 76, 82]. For separating a stiff, non-deformed spherical point contact, the DMT theory [65] predicts a necessary pull-off force FN,Z equivalent to the adhesion reaction force expressed by Eq. (17).

Compared with the stronger covalent, ionic, metallic or hydrogen bonds, these particle interactions are comparatively weak. From Eq. (18) the surface tension is about σsls = 0.25–50 mJ/m2 or the Hamaker constant according to the Lifshitz con-tinuum theory amounts to CH,sls = (0.2–40)⋅10–20 J [70, 75]. Notice here, the parti-cle interactions depend greatly on the applied load, which is experimentally con-firmed by atomic force microscopy [78, 79, 98].

A balance of stored elastic energy, mechanical potential energy and surface en-ergy delivers the contact radius of the two spheres [51], expressed here with a constant adhesion force FH,JKR from Eq. (23):

( )3 21 2K N H JKR H JKR N H JKR

32

2,

, , ,*

rr F F F F F

E

⋅= ⋅ + + ⋅ ⋅ +

⋅ (21)

Eq. (21) indicates a contact radius enhancement with increasing work of adhe-sion. The contact force-displacement relation is obtained from Eqs. (6) and (21) and can be compared with the Hertz relation Eq. (7) by the curves in Fig. 2 marked with Hertz and JKR:

3 3H,JKRN 1 2 K 1 2 K

423 3

**

, ,

E FEF r h r h⋅ ⋅⋅= ⋅ ⋅ − ⋅ ⋅ (22)

For small contact deformation the so-called JKR limit [51] is half of the con-stant adhesion force FH,JKR. This JKR model can be applied for higher bond strengths ΦT > 5 [76, 82–84]. It is valid for comparatively larger and softer parti-cles than the DMT model predicts [115]:

H JKRN Z JKR sls 1 23

2,

, , ,

FF rπ σ= = ⋅ ⋅ ⋅ (23)

Thus the contact radius for zero load FN = 0,

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Figure 2. Characteristic particle contact deformation. (c) Elastic–plastic compression [148]. The dominant linear elastic–plastic deformation range between pressure levels of powder mechanics [4] is demonstrated here. If the maximum pressure in the contact center reaches the micro-yield strength pmax = pf at the yield point Y then the contact starts with plastic yielding which is intensified by mo-bile adsorption layers. Next, the combined elastic–plastic yield boundary of the partial plate–plate contact is achieved as given in Eq. (58). This displacement is expressed with the annular elastic Ael (thickness rK,el) and circular plastic Apl (radius rK,pl) contact area, shown below panel c.


1 2 H JKR3K 0

3 , ,, *

r Fr

E

⋅ ⋅= (24)

is reduced to the pull-off contact radius, i.e.,

3K pull off K 0 4=, ,r r /- . (25)

Additionally, with applying an increasing tangential force FT, the contact radius rK is reduced by the last term within the square root:

2

3 21 2 TK N H JKR H JKR N H JKR

32

2 4

*,

, , ,* *

r F Er F F F F F

E G

� �⋅ ⋅� �= ⋅ + + ⋅ ⋅ + −� �⋅ ⋅� �

, (26)

When the square root in Eq. (26) disappears to zero, a critical value FT,crit is ob-tained, the so-called “peeling” of contact surfaces [64, 88]:

( )2

H JKR N H JKRT crit

22

*, ,

, *

F F F GF

E

⋅ ⋅ + ⋅= ⋅ (27)

An effective or net normal force (FN + FH,JKR) remains additionally in the con-tact [54]. Considering FT > FT,crit, i.e., contact failure by sliding (see Mindlin

[63]), the tangential force limit is expressed as ( )T i N H0tanF F Fϕ= ⋅ + . The ad-

hesion force FH0 (index H0) is constant during contact failure and the coefficient (or angle) of internal friction i itanµ ϕ= is also assumed to be constant for a

multi-asperity contact [50, 81, 82]. This constant friction was often confirmed for rough surfaces in both elastic and plastic regimes [50, 81, 84], but not for a sin-gle-asperity contact with nonlinear dependence of friction force on normal load [82, 84].

Rearranging Eq. (26), the extended contact force–displacement relation shows a reduction of the Hertz (first square-root) and JKR contributions to normal load FN which is needed to obtain a given displacement hK:

2

3 3H JKR TN 1 2 K 1 2 K

423 3 4

* **,

, , *

E F F EEF r h r hG

⋅ ⋅ ⋅⋅= ⋅ ⋅ − ⋅ ⋅ −⋅

(28)

However in terms of small particles (d < 10 µm), the increase of contact area with elastic deformation does not lead to a significant increase of attractive adhe-sion forces because of a practically too small magnitude of van der Waals energy of adhesion (Eq. (18)). The reversible elastic repulsion restitutes always the initial contact configuration during unloading.

Consequently, the increase of adhesion by compression, e.g., forming a snow ball, the well-known cohesive consolidation of a powder or the particle interac-

J. Tomas 16

tion and remaining strength after tabletting must be influenced by irreversible contact deformations, which are shown for a small stress level in a powder bulk in Figs 2 and 3.

If the maximum pressure pmax = pf in the center of the contact circle reaches the micro-yield strength, the contact starts with irreversible plastic yielding (index f). From Eqs. (2) and (5) the transition radius rK,f and from Eq. (6) the center ap-proach hK,f are calculated as:

1 2 fK f

,, *

r pr

E

π ⋅ ⋅= (29)

2 2

1 2 fK f 2

,, *

r ph

E

π ⋅ ⋅= (30)

Figure 2 demonstrates the dominant irreversible deformation over a wide range of contact forces. This transition point Y for plastic yielding is essentially shifted towards smaller normal stresses because of particle adhesion influence.

Rumpf et al. [62] and Molerus [13, 14] introduced this philosophy in powder mechanics and the JKR theory was the basis of adhesion mechanics [58, 67, 76, 85, 86, 90].

2.1.3. Perfect plastic and viscoplastic contact displacement Actually, assuming perfect contact plasticity, one can neglect the surface deformation outside of the contact zone and obtain with the following geometrical relation of a sphere

( )22 2 2K 1 1 K 1 1 K 1 K 1 1 K 12, , , ,r r r h r h h d h= − − = ⋅ ⋅ − ≈ ⋅ (31)

the total particle center approach of the two spheres:

2 2 2K K K

K K 1 K 21 2 1 22, ,

,

r r rh h h

d d r= + = + = ⋅ (32)

Because of this, a linear force–displacement relation is found for small spheri-cal particle contacts. The repulsive force as a resistance against plastic deforma-tion is given as:

N pl f K 1 2 f K, ,F p A d p hπ= ⋅ = ⋅ ⋅ ⋅ (33)

Thus, the contact stiffness is constant for perfect plastic yielding behavior, but decreases with smaller particle diameter d1,2 especially for cohesive fine powders and nanoparticles:

NN pl 1 2 f

K

dd, ,

Fk d p

hπ= = ⋅ ⋅ (34)


Figure 3. Characteristic particle contact deformation. (d) Elastic unloading and reloading with dis-sipation (titania) [148]. After unloading U – E the contact recovers elastically in the compression mode and remains with a perfect plastic displacement hK,E. Below point E on the axis the tension mode begins. Between the points U – E – A the contact recovers elastically according to Eq. (64) to a displacement hK,A. The reloading curve runs from point A to U to the displacement hK,U, Eq. (65).

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Additionally, the rate-dependent, perfect viscoplastic deformation (at the point

of yielding) expressed by contact viscosity ηK times indentation rate Kh� is as-

sumed to be equivalent to yield strength pf multiplied by indentation height in-crement hK

f K K Kp h hη⋅ = ⋅ � (35)

and one obtains again a linear model regarding strain rate:

N vis K K 1 2 K K, ,F A d hη π η= ⋅ = ⋅ ⋅ ⋅ �� (36)

An attractive viscous force is observed, e.g., for capillary numbers

K K lgCa 1h /η σ= ⋅ >� when comparatively strong bonds of (low-viscous) liquid

bridges are extended with negative velocity K–h� [71–73].

Consequently, the particle material parameters: contact micro-yield strength pf and viscosity ηK are measures of irreversible particle contact stiffness or softness. Both plastic and viscous contact yield effects were intensified by mobile adsorp-tion layers on the surfaces. The sum of deformation increments results in the en-ergy dissipation. For larger particle contact areas AK, the conventional linear elas-tic and constant plastic behavior is expected.

Now, what are the consequences of small contact flattening with respect to a varying, i.e., load or pre-history-dependent adhesion?

2.2. Particle contact consolidation by varying adhesion force

Krupp [49] and Sperling [48, 56] developed a model for the increase of adhesion force FH (index H) of the contact. This considerable effect is called here as “con-solidation” and is expressed as the sum of adhesion force FH0 according to Eq. (17) plus an attractive/repulsive force contribution due to irreversible plastic flat-tening of the spheres (pf is the repulsive “microhardness” or micro-yield strength of the softer contact material of the two particles, σss/a0 is the attractive contact pressure, index ss represents solid–vacuum–solid interaction):

ssH 1 2 ss

0 f

24 1,F r

a pσ

π σ⋅� �= ⋅ ⋅ ⋅ ⋅ +� �⋅� �

(37)

Dahneke [52] modified this adhesion model by the van der Waals force without any contact deformation FH0 plus an attractive van der Waals pressure (force per unit surface) pVdW contribution due to partially increasing flattening of the spheres which form a circular contact area AK (CH is the Hamaker constant based on inter-acting molecule pair additivity [69, 75]):


H 1 2 KH H0 K VdW 2

00

21

6,C r h

F F A paa

⋅ ⋅� �= + ⋅ = ⋅ +� �⋅ � � (38)

The distance a0 denotes a characteristic adhesion separation. If stiff molecular interactions are provided (no compression of electron sheath), this separation a0 was assumed to be constant during contact loading. By addition the elastic repul-sion of the solid material according to Hertz, Eq. (7), to this attraction force, Eq. (38), and by deriving the total force Ftot with respect to hK, the maximum adhesion force was obtained as absolute value

2

H 1 2 H 1 2H max 2 2 7

0 0

21

6 27

, ,, *

C r C rF

a E a

� �⋅ ⋅ ⋅� �= ⋅ +� �⋅ ⋅ ⋅� �

, (39)

which occurs at the center approach of the spheres [52]:

2H 1 2

K max 2 609

,, *

C rh

E a

⋅=

⋅ ⋅ (40)

But as mentioned before, this increase of contact area with elastic deformation does not lead to a significant increase of attractive adhesion force. The reversible elastic repulsion restitutes always the initial contact configuration. The practical experience with the mechanical behavior of fine powders shows that an increase of adhesion force is influenced by an irreversible or “frozen” contact flattening which depends on the external force FN [57].

Generally, if this external compressive normal force FN is acting at a single soft contact of two isotropic, stiff, smooth, mono-disperse spheres the previous contact point is deformed to a contact area, Fig. 1a to Fig. 2c, and the adhesion force be-tween these two partners increases, see in Fig. 3 the so-called “adhesion bound-ary” for incipient contact detachment. During this surface stressing the rigid parti-cle is not so much deformed that it undergoes a certain change of the particle shape. In contrast, soft particle matter such as biological cells or macromolecular organic material do not behave so.

For soft contacts Rumpf et al. [62] have developed a constitutive model ap-proach to describe the linear increase of adhesion force FH, mainly for plastic con-tact deformation:

( )VdW VdWH H0 N p H0 p N

f f1 1� �= + ⋅ + ⋅ = + ⋅ + ⋅� ��

p pF F F F F

p pκ κ (41)

With analogous prerequisites and derivation, Molerus [14] obtained an equiva-lent expression:

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VdWH H0 N H0 p N

f

pF F F F F

pκ= + ⋅ = + ⋅ (42)

The adhesion force FH0 without additional consolidation (FN = 0) can be ap-proached as a single rigid sphere–sphere contact (Fig. 1a). But, if this particle contact is soft enough the contact is flattened by an external normal force FN to a plate–plate contact (Fig. 2c). The coefficient κp describes a dimensionless ratio of attractive van der Waals pressure pVdW for a plate–plate model, Eq. (19), to repul-sive particle micro-hardness pf which is temperature sensitive:

H,slsVdWp 3

f 0 f6 =

= =⋅ ⋅ ⋅F

Cpp

π a pκ (43)

This is referred to here as a plastic repulsion coefficient. The Hamaker constant CH,sls for solid–liquid–solid interaction (index sls) according to Lifshitz’ theory [70] is related to continuous media which depends on their permittivities (dielec-tric constants) and refractive indices [75]. The characteristic adhesion separation for a direct contact is of a molecular scale (atomic center-to-center distance) and can be estimated for a molecular force equilibrium (a = aF=0) or interaction poten-tial minimum [75, 76, 91]. Its magnitude is about aF=0 ≈ 0.3–0.4 nm. This separa-tion depends mainly on the properties of liquid-equivalent packed adsorbed water layers. This particle contact behavior is influenced by mobile adsorption layers due to molecular rearrangement. The minimum separation aF=0 is assumed to be constant during loading and unloading for technologically relevant powder pres-sures σ < 100 kPa (Fig. 2c).

For a very hard contact this plastic repulsion coefficient is infinitely small, i.e., κp ≈ 0, and for a soft contact κp → 1.

If the contact circle radius rK is small compared to the particle diameter d, the elastic and plastic contact displacements can be combined and expressed with the annular elastic Ael and circular plastic Apl contact area ratio [57]:

VdWH H0 N

elf

pl

213

pF F F

Ap

A

= + ⋅� �

⋅ + ⋅� ��

(44)

For a perfect plastic contact displacement Ael → 0 and one obtains again Eq. (42):

H H0 p NF F Fκ≈ + ⋅ (45)

This linear enhancement of adhesion force FH with increasing pre-consolidation force FN, Eqs. (41), (42) and (45), was experimentally confirmed for micrometer sized particles, e.g., by Schütz [94, 95] (κp = 0.3 for limestone) and Newton [96] (κp = 0.333 for poly(ethylene glycol), κp = 0.076 for starch, κp =


0.017 for lactose, κp = 0.016 for CaCO3) with centrifuge tests [92] as well as by Singh et al. [97] (κp = 0.12 for poly(methylmethacrylate), κp ≈ 0 for very hard sapphire, α-Al2O3) with an Atomic Force Microscope (AFM). The two methods are compared with rigid and rough glass spheres (d = 0.1–10 µm), without any contact deformation, by Hoffmann et al. [98]. Additionally, using the isostatic tensile strength σ0 determined by powder shear tests [91, 122, 147, 149, 151], this adhesion level is of the same order of magnitude as the average of centrifuge tests (see Spindler et al. [99]).

The enhancement of adhesion force FH due to pre-consolidation was confirmed by Tabor [30], Maugis [85, 86] and Visser [110]. Also, Maugis and Pollock [58] found that separation was always brittle (index br) with a small initial slope of pull-off force, dFN,Z,br/dFN (FN,Z,br ≈ – FH), for a comparatively small surface en-ergy σss of the rigid sphere–gold plate contact (index ss). In contrast, a pull-off

force FN,Z,br proportional to NF was obtained from the JKR theory [58] for the

full plastic range of high loading and brittle separation of the contact (Table 1):

* NN,Z,br ss 3

f

= − ⋅ ⋅⋅F

F Ep

σπ

(46)

Additionally, a load-dependent adhesion force was also experimentally con-firmed in wet environment of the particle contact by Butt and co-workers [78, 79] and Higashitani and co-workers [87] with AFM measurements.

The dominant plastic contact deformation of surface asperities during the chemical–mechanical polishing process of silicon wafers was also recognized, e.g., by Rimai and Busnaina [111] and Ahmadi and Xia [141]. These particle-surface contacts and, consequently, asperity stressing by simultaneous normal pressure and shearing, contact deformation, microcrack initiation and propaga-tion, and microfracture of brittle silicon asperity peaks affect directly the polish-ing performance. Thus the Coulomb friction becomes dominant also in a wet en-vironment.

2.3. Variation in adhesion due to non-elastic contact consolidation

2.3.1. Elastic–plastic force–displacement model All interparticle forces can be expressed in terms of a single potential function

i i i i( ) /= ±∂ ∂F U h h and thus are superposed. This is valid only for a conservative

system in which the work done by the force Fi versus distance hi is not dissipated as heat, but remains in the form of mechanical energy, simply in terms of irre-versible deformation, e.g., initiation of nanoscale distortions, dislocations or lat-tice stacking faults. The overall potential function may be written as the sum of the potential energies of a single contact i and all particle pairs j. Minimizing this potential function ij ij

i j

/ 0∂ ∂ =�� U h one obtains the potential-force balance.

J. Tomas 22

Thus, the elastic–plastic force–displacement models introduced by Schubert et al. [57], Eq. (44), and Thornton [60] Eq. (47)

( )N f 1,2 K K,f / 3= π⋅ ⋅ ⋅ −F p r h h (47)

should be supplemented here with a complete attractive force contribution due to contact flattening described before. Taking into account Eqs. (41), (42) and (44), the particle contact force equilibrium between attraction (-) and elastic plus, si-

multaneously, plastic repulsion (+) is given by ( *Kr represents the coordinate of

annular elastic contact area):

K

K,pl

2 2H0 VdW K N f K,pl

* * *el K K K

0

2 ( ) d

= = − − ⋅ ⋅ − + ⋅ ⋅

+ ⋅ ⋅ ⋅

�

�r

r

F F p r F p r

p r r r

π π

π (48)

3 / 222K,pl2 2 max K

N H0 VdW K f K,plK

21

3

� �� ⋅ ⋅ ⋅� �+ + ⋅ ⋅ = ⋅ ⋅ + ⋅ − � ��

rp rF F p r p r

rπ

π π (49)

At the yield point rK = rK,pl the maximum contact pressure reaches the yield strength pel = pf.

22

2 2 K fN H0 VdW K f K,pl

max

23

� �⋅ ⋅ � �� + + ⋅ ⋅ = ⋅ ⋅ + ⋅� ��

r pF F p r p r

pπ

π π (50)

Because of plastic yielding, a pressure higher than pf is absolutely not possible and thus, the fictitious contact pressure pmax is eliminated by Eq. (1):

22

K,pl2 2 KN H0 VdW K f K,pl 2

K

21

3

� �� ⋅ ⋅� �� + + ⋅ ⋅ = ⋅ ⋅ + ⋅ −

� ��

rrF F p r p r

r

ππ π (51)

Finally, the contact force equilibrium

2K,pl2

N H0 VdW K f K 2K

plf K

K

2 13 3

2 13 3

� �� + + ⋅ = ⋅ ⋅ ⋅ + ⋅� ��

� �= ⋅ ⋅ + ⋅� �

� ��

rF F p A p r

r

Ap A

A

π

(52)

and the total contact area AK are obtained:


N H0K

plf VdW

K

2 13 3

+=

� �⋅ + ⋅ −� ��

F FA

Ap p

A

(53)

Next, the elastic–plastic contact area coefficient κA is introduced. This dimen-sionless coefficient represents the ratio of plastic particle contact deformation area Apl to total contact deformation area K pl el= +A A A and includes a certain elastic

displacement:

plA

K

2 13 3

= + ⋅A

Aκ (54)

The solely elastic contact deformation Apl = 0, κA = 2/3, has only minor rele-vance for cohesive powders in loading (Fig. 2), but for the complete plastic con-tact deformation (Apl = AK) the coefficient κA = 1 is obtained.

From Eqs. (43), (53) and (54) the sum of contact normal forces is obtained as:

( )2N H0 K f A p+ = ⋅ ⋅ ⋅ −F F r pπ κ κ (55)

From Eq. (5) the transition radius of elastic-plastic model rK,f,el-pl (index el-pl) and from Eq. (6) the particle center approach of the two particles hK,f,el-pl are cal-culated as:

( )1,2 f A p

K,f ,el pl *

3

2−

⋅ ⋅ ⋅ ⋅ −=

⋅

r pr

E

π κ κ (56)

( ) 22 2

1,2 f A pK,f ,el pl 2*

9

4−

⋅ ⋅ ⋅ ⋅ −=

⋅

r ph

E

π κ κ (57)

Checking this model, Eq. (56), with pure elastic contact deformation, i.e., κp → 0 and κA = 2/3, the elastic transition radius rK,f, Eq. (29), is also obtained. For ex-ample, nanodisperse titania particles (d50,3 = 610 nm is the median diameter on mass basis (index 3), E = 50 kN/mm2 modulus of elasticity, ν = 0.28 Poisson ra-tio, pf = 400 N/mm2 micro-yield strength, κA ≈ 5/6 contact area ratio, κp = 0.44 plastic repulsion coefficient) a contact radius of rK,f,el-pl = 2.1 nm and, from Eq. (57), a homeopathic center approach of only hK,f,el-pl = 0.03 nm are obtained. This is a very small indentation calculated, in principle, by means of a continuum ap-proach. The contact deformation is equivalent to a microscopic force FN = 2.1 nN or to a small macroscopic pressure level of about σ = 1.4 kPa (porosity ε = 0.8) in powder handling and processing.

J. Tomas 24

Introducing the particle center approach of the two particles Eq. (6) in Eq. (55), a very useful linear force–displacement model approach is obtained again for κA ≈ constant:

( )N H0 1,2 f A p K+ = ⋅ ⋅ ⋅ − ⋅F F r p hπ κ κ (58)

But if one considers the contact area ratio of Eq. (63), a slightly nonlinear (pro-gressively increasing) curve is obtained. Using the elastic–plastic contact consoli-dation coefficient κ due to definition (Eq. (71)) one can also write:

1,2 f AN H0 K1

⋅ ⋅ ⋅+ = ⋅+

r pF F h

π κκ (59)

The curve of this model is shown in Fig. 2 for titania powder which was recalculated from material data and shear test data [147, 149]. The slope of this plastic curve is a measure of irreversible particle contact stiffness or softness, Eq. (34). Because of particle adhesion impact, the transition point for plastic yielding Y is shifted to the left compared with the rough calculation of the displacement limit hK,f by Eq. (30).

The previous contact model may be supplemented by viscoplastic stress-strain behavior, i.e., strain-rate dependence on initial yield stress. For elastic–visco-plastic contact, one obtains deformation with Eqs. (36) and (58) (κA ≈ constant):

( )N H0 1,2 K A p,t K+ = ⋅ ⋅ ⋅ − ⋅ �F F r hπ η κ κ (60)

A dimensionless viscoplastic contact repulsion coefficient κp,t is introduced as the ratio of the van der Waals attraction to viscoplastic repulsion effects which are additionally acting in the contact after attaining the maximum pressure for yield-ing.

VdWp,t

K K

=⋅ �

p

hκ

η (61)

The consequences for the variation in adhesion force are discussed in Section 2.3.3 [147].

2.3.2. Unloading and reloading hysteresis and contact detachment Between the points U – E (see Fig. 3), the contact recovers elastically along an extended Hertzian parabolic curve, Eq. (7), down to the perfect plastic displace-ment, hK,E, obtained in combination with Eq. (58):

23K,E K,U K,f K,U= − ⋅h h h h (62)


Thus, the contact area ratio κA is expressed more in detail with Eqs. (6) and (54) for elastic κA = 2/3 and perfect plastic contact deformation, κA = 1 if hK,U → ∞:

K,E K,f3

AK,U K,U

2 113 3 3

= + = − ⋅⋅h h

h hκ (63)

Beyond point E to point A, the same curve runs down to the intersection with the adhesion boundary, Eq. (67), to the displacement hK,A:

( )3*N,unload 1,2 K K,A H,A

23

= ⋅ ⋅ ⋅ − −F E r h h F (64)

Consequently, the reloading runs along the symmetric curve

( )3*N,reload 1,2 K,U K N,U

23

= − ⋅ ⋅ ⋅ − +F E r h h F (65)

from point A to point U to the displacement hK,U as well (Fig. 3). The displace-ment hK,A at point A of contact detachment is calculated from Eqs. (57), (58), (64) and (67) as an implied function (index (0) for the beginning of iterations) of the displacement history point hK,U:

( )23

K,A,(1) K,U K,f ,el pl K,U K,A,(0)= − ⋅ + ⋅h h h h hκ- (66)

If one replaces FN in Eq. (72) (see Section 2.3.4), by the normal force–displacement relation, Eq. (58), additionally one obtains a plausible adhesion force–displacement relation which shows the increased pull-off force level after contact flattening, hK = hK,A compared with Eq. (38) and point A in the diagram of Fig. 4:

H,A H0 1,2 VdW K,A= + ⋅ ⋅ ⋅F F r p hπ (67)

The unloading and reloading hysteresis for an adhesion contact takes place be-tween the two characteristic straight-lines for compression, the elastic–plastic yield boundary Eq. (58), and for tension, the remaining adhesion (pull-off) boundary Eq. (67) and Fig. 3.

At this so-called adhesion (failure) boundary the contact microplates fail and detach with the increasing distance 0 K,A K== + −Fa a h h . The actual particle sepa-

ration a can be used by a long-range hyperbolic adhesion force curve 3N,Z

−∝F a

with the van der Waals pressure pVdW as given in Eq. (19) and the displacement hK,A for incipient contact detachment by Eq. (66):

J. Tomas 26

Figure 4. Characteristic particle contact deformation. (e) Contact detachment [148]. Again, if one applies a certain pull-off force FN,Z = –FH,A as given in Eq. (67) but here negative, the adhesion boundary line at failure point A is reached and the contact plates fail and detach with the increasing distance 0 K A K== + −F ,a a h h . This actual particle separation is considered for the calculation by a

hyperbolic adhesion force curve FN,Z = 3H A,–F a−∝ of the plate–plate model Eq. (68).


Figure 5. Characteristic particle contact deformation. The complete survey of loading, unloading, reloading, dissipation and detachment behaviors of titania [148]. This hysteresis behavior could be shifted along the elastic–plastic boundary and depends on the pre-loading, or in other words, pre-consolidation level. Thus, the variation in adhesion forces between particles depends directly on this frozen irreversible deformation, the so-called contact pre-consolidation history.

H0 1,2 VdW K,AN,Z K 3

K,A K

0 0

( )

1= =

+ ⋅ ⋅ ⋅= −

� �+ −� �

� �F F

F r p hF h

h ha a

π (68)

These generalized functions in Fig. 3 for the combination of elastic-plastic, ad-hesion and dissipative force–displacement behaviors of a spherical particle con-tact were derived on the basis of the theories of Krupp [49], Molerus [13], Maugis

J. Tomas 28

[58], Sadd [55] and, especially, Thornton [53, 60]. A complete survey of loading, unloading, reloading, dissipation and detachment behaviors of titania is shown in Fig. 5 as a combination of Fig. 1a to Fig. 4e. This approach may be expressed here in terms of engineering mechanics of macroscopic continua [1, 2] as the his-tory-dependent contact behavior.

2.3.3. Viscoplastic contact behavior and time dependent consolidation An elastic-plastic contact may be additionally deformed during the indentation time, e.g., by viscoplastic flow (Section 2.1.3). Thus, the adhesion force increases with interaction time [32, 49, 77, 128]. This time-dependent consolidation behav-ior (index t) of particle contacts in a powder bulk was previously described by a parallel series (summation) of adhesion forces, see Table 1, last line marked with Tomas [122–125, 146–149]. This method refers more to incipient sintering or contact fusion of a thermally-sensitive particle material [62] without interstitial adsorption layers. This micro-process is very temperature sensitive [122, 124, 125, 146].

Additionally, the increasing adhesion may be considered in terms of a sequence of rheological models as the sum of resistances due to plastic and viscoplastic re-pulsion κp + κp,t, line 5 in Table 2. Hence the repulsion effect of “cold” viscous flow of comparatively strongly-bonded adsorption layers on the particle surface is taken into consideration. This rheological model is valid only for a short-term in-

Table 2. Material parameters for characteristic adhesion force functions FH(FN) in Fig. 8

Instantaneous contact consolidation

Time-dependent consolidation

Constitutive model of contact deformation

plastic viscoplastic

Repulsion coefficient H slsVdW

p 3f 0 f6 =

= =⋅ ⋅ ⋅

,

F

Cpp a p

κπ

VdWp t

K,

ptκ η= ⋅

Constitutive models of combined contact deformation

elastic–plastic elastic–plastic and viscoplastic

Contact area ratio

( )pl

Apl el

23 3

A

A Aκ = +

⋅ + ( )

pl visA t

pl vis el

23 3

,

A A

A A Aκ

+= +

⋅ + +

Contact consolidation coefficient

p

A p

κκ κ κ= − p p t

visA t p p t

,

, ,

κ κκ κ κ κ

+= − −

Intersection with FN-axis (abscissa)

( )1 2N Z 0 f

H sls

,, F r

,

F a h p

f C

π =≈ − ⋅ ⋅ ⋅

≠

( )

1 20 f

N Z totf K

H sls

1=⋅ ⋅ ⋅

≈ − + ⋅

≠

,F r

, ,

,

a h pF

p t /

f C

πη


Figure 6. Characteristic elastic–plastic, viscoelastic–viscoplastic particle contact deformations (tita-nia, primary particles d = 20–300 nm, surface diameter dS = 200 nm, median particle diameter d50,3 = 610 nm, specific surface area AS,m = 12 m2/g, solid density ρs = 3870 kg/m3, surface moisture XW = 0.4%, temperature θ = 20°C, loading time t = 24 h). The material data, modulus of elasticity E = 50 kN/mm2, modulus of relaxation E∞ = 25 kN/mm2, relaxation time trelax = 24 h, plastic micro-yield strength pf = 400 N/mm2, contact viscosity ηK = 1.8·1014 Pa·s, Poisson ratio ν = 0.28, Hamaker constant CH,sls = 12.6·10-20 J, equilibrium separation for dipole interaction aF=0 = 0.336 nm, contact area ratio κA = 5/6 are assumed as appropriate for the characteristic contact properties. The plastic repulsion coefficient κp = 0.44 and viscoplastic repulsion coefficient κp,t = 0.09 are recalculated from shear-test data in a powder continuum [147, 149].

J. Tomas 30

Figure 7. Constitutive models of contact deformation of smooth spherical particles in normal direc-tion without (only compression +) and with adhesion (tension –). The basic models for elastic be-havior were derived by Hertz [41], for constant adhesion by Yang [61], for constant adhesion by Johnson et al. [51], for plastic behavior by Thornton and Ning [60] and Walton and Braun [59], and for plasticity with variation in adhesion by Molerus [13] and Schubert et al. [57]. This has been ex-panded stepwise to include nonlinear plastic contact hardening and softening equivalent to shear-thickening and shear-thinning in suspension rheology [91]. Energy dissipation was considered by Sadd et al. [55] and time-dependent viscoplasticity by Rumpf et al. [62]. Considering all these theo-ries, one obtains a general contact model for time- and rate-dependent viscoelastic, plastic, vis-coplastic, adhesion and dissipative behaviors [91, 146–148, 151].


dentation ( )K f/< ⋅t pη κ , e.g., t < 5 d for the titania used as a very cohesive

powder (specific surface area AS,m = 12 m2/g, with a certain water adsorption ca-pacity). All the material parameters are collected in Table 2.

A viscoelastic relaxation in the particle contact may be added as a time-dependent function of the average modulus of elasticity E*, Yang [61] and Krupp [49] (trelax is the characteristic relaxation time):

( )relax* * * *0

1 1 1 1 exp( ) ( 0)∞ ∞

� �� = + − ⋅ −� �→ ∞ =� �

t tE E t E t E

(69)

The slopes of the elastic–plastic, viscoelastic–viscoplastic yield and adhesion boundaries as well as the unloading and reloading curves, which include a certain relaxation effect, are influenced by the increasing softness or compliance of the spherical particle contact with loading time (Fig. 6). This model system includes all the essential constitutive functions of the authors named before [41, 55, 57, 60, 61]. A survey of the essential contact force-displacement models is given in Fig. 7 and Table 1.

Obviously, contact deformation and adhesion forces are stochastically distrib-uted material functions. Usually one may focus here only on the characteristic or averaged values of these constitutive functions.

2.3.4. Adhesion force model Starting with all these force-displacement functions one turns to an adhesion and normal force correlation to find out the physical basis of strength-stress relations in continuum mechanics [13, 14, 122, 149]. Replacing the contact area in Eq. (38), the following force–force relation is directly obtained:

VdW H0 NH H0 VdW K H0

f pl VdW

K f

2 13 3

+= + ⋅ = + ⋅

+ ⋅ −

p F FF F p A F

p A pA p

(70)

Therefore, with a so-called elastic–plastic contact consolidation coefficient κ,

p

A p= −

κκ κ κ (71)

a linear model for the adhesion force FH as function of normal force FN is ob-tained (Fig. 8):

( )pAH H0 N H0 N

A p A p1= ⋅ + ⋅ = + ⋅ + ⋅− −F F F F F

κκκ κκ κ κ κ (72)

The dimensionless strain characteristic κ is given by the slope of adhesion force FH which is influenced by predominant plastic contact failure. It is a meas-

J. Tomas 32

ure of irreversible particle contact stiffness or softness. A shallow slope desig-nates a low adhesion level FH ≈ FH0 because of stiff particle contacts, but a large slope means soft contacts, or consequently, a cohesive powder flow behavior [91, 147, 149]. The contact flattening may be additionally dependent on time or dis-placement rate (Section 2.1.3). Thus, the contact reacts softer and, consequently, the adhesion level is higher than before. This new adhesion force slope κvis is modified by the viscoplastic contact repulsion coefficient κp,t, which includes cer-tain viscoplastic microflow at the contact (Table 2 and Fig. 8),

( )

p p tA tH tot H0 N

A t p p t A t p p t

vis H0 vis N1

+= ⋅ + ⋅− − − −

= + ⋅ + ⋅

,,,

, , , ,

F F F

F F

κ κκκ κ κ κ κ κ

κ κ

(73)

with the so-called total viscoplastic contact consolidation coefficient κvis that in-cludes the elastic-plastic κp and the viscoplastic contributions κp,t of contact flat-tening,

Figure 8. Particle contact forces for titania powder (median particle diameter d50,3 = 610 nm, spe-cific surface area AS,m = 12 m2/g, surface moisture XW = 0.4%, temperature = 20°C) according to the linear model Eq. (72), non-linear model Eq. (79) for instantaneous consolidation t = 0 and the linear model for time consolidation t = 24 h (Eq. (73)). The powder surface moisture XW = 0.4% is accu-rately analyzed with Karl–Fischer titration. This is equivalent to an idealized mono-molecular ad-sorption layer being in equilibrium with an ambient air temperature of 20°C and 50% humidity.


p p,tvis

A,t p p,t

+= − −

κ κκ κ κ κ (74)

Eqs. (72) and (73) consider also the flattening response of soft particle con-tacts at normal force FN = 0 caused by the adhesion force κ⋅FH0 (Krupp [56]) and κvis⋅FH0. Hence, the adhesion force FH0 represents the sphere–sphere contact with-out any contact deformation at minimum particle–surface separation aF=0. This initial adhesion force FH0 may also include a characteristic nanometer-sized height or radius of a rigid spherical asperity

1,2=0 1,2< << F ra h r with its center lo-

cated at an average radius of the spherical particle, Krupp [49], Rumpf et al. [33] and Schubert et al. [34]:

( )1,2 1,2 1,2

1,2

H,sls 1,2 H,sls

H0 2 2 20 0

0

/1

6 61 2 /= ==

� �⋅ ⋅� �

= ⋅ + ≈� �⋅ ⋅� �+ ⋅

� ��

r r r

F Fr F

C h r h C hF

a ah a (75)

This rigid adhesion force contribution FH0, see also the fundamentals [44, 69] and supplements [32, 49, 56, 67, 98–112], is valid only for perfect stiff contacts. Additionally, for mono-sized spheres 1,24= ⋅d r an averaged asperity height

1,22= ⋅r rh h can be used with ( )

1,2 1 2

11 / 1 /

−= +r r rh h h . When the asperity height is

not too far from the averaged sphere radius 1,2 1,2< rh r , then the adhesion force can

be calculated as, see Hoffmann (index Ho) [98]:

( )

( )

1 2 1 2

1 2

1 2

1 2

1 2

H sls 1 2

2 2H0 Ho1 20

0

H sls

2

0 1 2

116 1 2

6 1

==

=

� �⋅ � �= ⋅ +� �+⋅ + ⋅� ��

⋅≈

⋅ ⋅ +

, ,

,

,

,

,

, r , r

,r ,F

r F

, r

F r ,

C h r / hF

h / ra h / a

C h

a h / r

(76)

If the radius of the roughness exceeds the minimum separation of the sphere–plate system in order of magnitudes (hr >> aF=0), the contribution of the plate, second term in Eq. (76), can be neglected and the adhesion force may be de-scribed as the sphere-sphere contact [98].

Rabinovich and co-workers [113–115] have used the root mean square (RMS) roughness from AFM measurements and the average peak-to-peak distance be-tween these asperities λr to calculate the interaction between a smooth sphere and a surface with nanoscale roughness profile (index Ra):

J. Tomas 34

( )

H,sls 1,2H0,Ra 2 2 2

0 1,2 0

1 16 1 58.14 RMS / 1 1.817 RMS /= =

� �⋅� �= ⋅ +� �⋅ + ⋅ ⋅ + ⋅� �F r F

C rF

a r aλ (77)

The first term in brackets represents the contact interaction of the particle with an asperity and the second term accounts for the non-contact interaction of the particle with an average surface plane. This approach describes stiff nanoscale roughness as caps of asperities with their centers located far below the surface. For example, RMS roughness of only 1 or 2 nm is significant enough to reduce the theoretical adhesion force FH0 by an order of magnitude or more [115]. Greenwood [50, 80, 81] and Johnson [67] described the elastic and plastic defor-mations of random surface asperities of contacts by the standard deviation of roughness and mean pressure.

The intersection of function (72) with abscissa (FH = 0) in the negative of con-solidation force FN (Fig. 8), is surprisingly independent of the Hamaker constant CH,sls:

( )1 2

1 2

1 2

1 2

1 2

2N Z 0 f A

0

0 f

11 2

=

=

=

� �� = − ⋅ ⋅ ⋅ ⋅ ⋅ +� �

+ ⋅� ��

≈ − ⋅ ⋅ ⋅

,

,

,

,

, r

, F r

r F

F r

r / hF a h p

h / a

a h p

π κ

π

(78)

This minimum normal (tensile or pull-off) force limit FN,Z for nearly brittle contact failure combines the influences of the particle contact hardness pf ≈ (3–15)⋅σf (σf = yield strength in tension, details in Ghadiri [117]) for a confined plastic micro-stress field in indentation [116] and the particle separation distribu-tion, which is characterized here by the mean particle roughness height

1,2rh , and

the molecular center separation aF=0. Obviously, this value characterizes also the contact softness with respect to a small asperity height hr as well, see Eq. (34). This elastic–plastic model (Eq. (72)) can be interpreted as a general linear consti-tutive contact model concerning loading pre-history-dependent particle adhesion, i.e., linear in forces and stresses, but non-linear regarding material characteris-tics.

But if one eliminates the center approach hK of the loading and unloading func-tions, Eqs. (58) and (64), an implied non-linear function between the contact pull-off force FH,A = – FN,Z at the detachment point A is obtained for the normal force at the unloading point FN = FN,U:


( )

( )H A (1) H0 N H0

2 3

2 H A (0) H0N H021 2 p f

N H01 2

31

2

= + ⋅ +

� �−⋅ + � �− ⋅ ⋅ ⋅ ⋅ ⋅ +� �� +⋅ ⋅� � � �

, ,

/

, ,*,

,

F F F F

F FF F r p

F Fr E

κ

π κ (79)

This unloading point U is stored in the memory of the contact as pre-consolidation history. This general non-linear adhesion model (dashed curve in Fig. 8) implies the dimensionless, elastic-plastic contact consolidation coefficient κ and, additionally, the influence of adhesion, stiffness, average particle radius r1,2, average modulus of elasticity E* in the last term of the equation. The slope of the adhesion force is reduced with increasing radius of surface curvature r1,2. Generally, the linearised adhesion force (Eq. (72)) is used first to demonstrate comfortably the correlation between the adhesion forces of microscopic particles and the macroscopic stresses in powders [91, 146, 147]. Additionally, one can ob-tain a direct correlation between the micromechanical elastic-plastic particle con-tact consolidation and the macro-mechanical powder flowability expressed by the semi-empirical flow function ffc according to Jenike [4].

It should be pointed out here that the adhesion force level in Fig. 8 is approxi-mately 105–106 times the particle weight for fine and very cohesive particles. This means, in other words, that one has to apply these large values as acceleration ra-tios a/g with respect to gravity to separate these pre-consolidated contacts or to remove mechanically such adhered particles from surfaces.

For a moist particle packing, the liquid-bridge-bonding forces caused by capil-lary pressure of interstitial pores and surface tension contribution of the free liq-uid surface additionally determine the strength [118–122]. Attraction by capillary pressure and increasing van der Waals forces by contact flattening due to normal load (application of an external pressure) are also acting in particle contacts of compressed water-saturated filter cakes or wet-mass powders [91, 144, 145, 150].

2.4. Energy absorption in a contact with dissipative behavior

If one assumes a single elastic–plastic particle contact as a conservative mechani-cal system without heat dissipation, the energy absorption equals the lens-shaped area between the unloading and reloading curves from point U to A as shown in Fig. 3:

K,U K,U

K,A K,A

diss N,reload K K N,unload K K( ) d ( ) d= −� �

h h

h h

W F h h F h h (80)

from Eqs. (64), (67) for FH,A and (65), (58) for FN,U, one obtains finally:

J. Tomas 36

( )

( ) ( )

5

diss 1 2 K U K A

1 2 f A K U p K U K A K U K A

815⋅= − ⋅ −

� �+ ⋅ ⋅ ⋅ ⋅ − ⋅ − ⋅ −� �

*

, , ,

, , , , , ,

EW r h h

r p h h h h hπ κ κ (81)

Additionally, the specific or mass-related energy absorption includes the aver-

age particle mass 3P 1,2 s4 / 3= ⋅m πr ρ a characteristic contact number in the bulk

powder (coordination number k ≈ π/ε [13]) and the dissipative work m,diss =W

diss P/⋅k W m :

( ) ( )

5 2

K U K Am diss

s 1 2

f K U K A

2 A K U p K U K A1 2 s

20

3

32

−� �−= ⋅� �⋅ ⋅ � �

⋅ ⋅ ⋅ −� �+ ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅

/*

, ,,

,

, ,

, , ,

,

h hEWr

p h hh h h

r

ε ρ

πκ κ

ε ρ

(82)

A specific energy absorption of 3 to 85 µJ/g was dissipated during a single unloading–reloading cycle in the titania bulk powder with an average pressure of only σM,st = 2 to 18 kPa (or major principal stress σ1 = 4 to 33 kPa) [147, 149].

3. PARTICLE IMPACT AND CONTACT DISPLACEMENT RESPONSE

In a shear zone, when two particles (particle 2 is assumed to be fixed) come into contact and collide, the velocity of particle 1 is reduced gradually. Part of the ini-tial kinetic energy is radiated into both particles as elastic waves. Now the contact force reaches a maximum value (maximum de-acceleration) and the particle ve-locity is reduced to zero.

K

* 3 * 5el 1,2 K K 1,2 K

0

2 4d3 15

= ⋅ ⋅ = ⋅ ⋅�

h

W E r h h E r h (83)

With the particle mass 31,2 s 1,2

43

= ⋅ ⋅ ⋅m rρ π , the correlation between particle ve-

locity v1 and center approach hK is obtained:

5 2

2 K1

1 23

� �= ⋅� �⋅ ⋅ � �

/*

s ,

hEvrπ ρ (84)

In the recovery stage the stored elastic energy is released and converted into kinetic energy and the particle moves with the rebound velocity v1,R into the oppo-site direction.


The so-called impact number or coefficient of restitution 1,R 1ˆ ˆ/=e F F indicates

the impact force ratio of the contact decompression phase after impact and the contact compression phase during impact, e = 0 for perfect plastic, 0 < e < 1 for elastic–plastic, e = 1 for perfect elastic behavior, see examples in Refs. [29, 126, 130, 132]. Thus e2 < 1 characterizes the energy dissipation (Wdiss is the inelastic

deformation work of particle contact, 2kin,1 P 1 / 2= ⋅E m v is the kinetic energy of

particle 1 before impact):

2 kin,1 diss

kin,1

−=

E We

E (85)

In terms of a certain probability of particle adhesion inside of the contact zone a critical velocity (index H) as the stick/bounce criterion was derived by Thornton (index Th) [60] who used the JKR model:

1/ 32H,JKR H,JKR

1,H,Th 2P

1.871 3

d

� �⋅ ⋅� �= ⋅� �⋅� �

*

F Fv

m E (86)

For an impact velocity v1 > v1,H particle bounce occurs and the coefficient of restitution is obtained as [60]:

2 21,R 1,H

2 21 1

1= = −v v

ev v

(87)

Even if the impact velocity v1 is 10-times higher than the critical sticking veloc-ity v1,H,Th the coefficient of restitution is 0.995 [60].

But in terms of combined elastic–plastic deformation the kinetic energy is mainly dissipated. If one uses the center approach hK,f of Eq. (30) the critical im-pact velocity v1,f for incipient plastic yield (index f) is calculated from Eq. (84) as [131–133]:

2

f f1,f *

s3⋅� �= ⋅� � ⋅� �

p pv

E

πρ (88)

The critical velocity v1,H to stick or to adhere the particles with a plastic contact deformation was derived by Hiller (Index HL) [126]:

( )1/ 22

H,sls1,H,HL 2 2

0 s f

11

6=

−= ⋅ ⋅

⋅ ⋅ ⋅ ⋅F

e Cv

de a pπ ρ (89)

J. Tomas 38

Figure 9. Recalculated plastic contact deformation and sticking/bounce at central impact stressing using data from Fig. 6. Two particles approach with velocities v1 and v2, impact and the contact is elastic–plastically deformed (top panel). The inelastic deformation energy is dissipated into the con-tact. This is equivalent to the areas (gray tones) between the elastic–plastic boundary and adhesion boundary of the force–displacement lines which are obtained by integration (Eq. (91)). If the kinetic energy of these particles would be large enough, these particles can detach with rebound velocities v1R and v2R. The critical impact velocity for incipient yield of the contact is shown (Eq. (88)). Above this value, the two particles adhere or stick in practice, i.e., v1R = 0. From this, the critical impact ve-locity v1,H follows and is shown in the bottom panel versus particle center approach or displacement hK,U. The model of Hiller/Löffler predicts a constant velocity (Eq. (90)). However, practical experi-ence shows us that the faster the particles move and impact, the larger the contact displacement, and consequently, the higher the tendency to stick. This is demonstrated by the curve of Eq. (93) in the bottom panel versus displacement hK,U.


This can be rearranged if one uses the dimensionless plastic repulsion coeffi-cient κp according to Eq. (43) to obtain the following simple expression:

( )1/ 22

0 f1,H,HL p2

s

1 6=− ⋅

= ⋅ ⋅ ⋅Fe a p

vde

κ ρ (90)

Unfortunately, Eq. (90) does not include the increase of “soft” contact flatten-ing response hK by increasing particle impact velocity v1. Now this dominant en-ergy absorption Wdiss during particle impact stressing, beginning at any unloading point U, is considered approximately as a trapezium-shaped area between elastic–plastic yield boundary and adhesion boundary for the contact of particles 1 and 2 in the force–displacement diagram of Fig. 9. With the contribution of the work of adhesion WA to separate this contact from equilibrium separation aF=0 to infinity, the energy balance gives (AK is the contact area):

( )K U K U

K f K f

0

2 21 21 1 R N K K N Z K K

K VdW

( ) d ( ) d2

( ) d=

∞

⋅ − = + −

+ ⋅ −

� �

�

, ,

, ,

F

h h

,, ,

h h

a

mv v F h h F h h

A p a a

(91)

( )2 2 2 2f1 1,R A K,U K,f p K,U 02

s 1,2

3

4 =⋅ � �− = ⋅ ⋅ − + ⋅ ⋅

� �⋅ ⋅ F

pv v h h h a

rκ κ

ρ (92)

The difference in characteristic impact velocities results directly in a center ap-proach, hK,U, expressed by the unloading point U. The response of this contact displacement hK,U is a consolidation force, FN,U. Additionally, a certain pre-consolidation level, FN,U, in a shear zone may affect the sticking/bounce probabil-ity. If the rebound velocity v1,R = 0 the two particles will adhere. Consequently, the critical sticking velocity v1,H is obtained without any additional losses, e.g., due to elastic wave propagation:

( )2 2f1,H A K,U K,f p K,U 02

s 1,2

3

4 =⋅ � �= ⋅ ⋅ − + ⋅ ⋅

� �⋅ ⋅ F

pv h h h a

rκ κ

ρ (93)

For example using data from Refs. [147, 149], this critical sticking velocity lies between 0.2 and 1 m/s for titania, curve in the sticking velocity–displacement diagram in Fig. 9, which is equivalent to an average pressure level σM,st = 2 to 18 kPa (or major principal stress σ1 = 4 to 33 kPa) [151]. These calculation results of particle adhesion are in agreement with the practical experiences in powder han-dling and transportation, e.g., undesired powder blocking at conveyer transfer chutes. In terms of powder flow the behavior after multiple stressing of soft de-

J. Tomas 40

forming contacts in the nanoscale of center approach hK, may be described as “healing contacts”.

To demonstrate this enormous adhesion potential, 1-µm silica particles were completely removed from a 100-mesh woven metal screen (147 µm wide) with 40 m/s air velocity [127] and 32-µm glass beads from glass surface with more than 117 m/s [129]. Air velocities of 10 to 20 m/s were necessary to blow off about 50% of quartz particles (d = 5–15 µm) which had adhered to filter media after impact velocities of about 0.28 to 0.84 m/s [28, 29].

These fundamentals of particle adhesion dynamics may also be important to chemically clean silicon wafers [36, 134–141] or mechanical tool surfaces by jet pressures up to 2 MPa and CO2-ice particle velocities up to 280 m/s [135].

4. CONCLUSIONS

The models for elastic (Hertz, Huber, Cattaneo, Mindlin and Deresiewicz), elas-tic–adhesion (Derjaguin, Johnson), plastic–adhesion (Derjaguin, Krupp, Molerus, Johnson, Maugis and Pollock) contact deformation response of a single, normal or tangential loaded, isotropic, smooth contact of two spheres were discussed. The force–displacement behaviors of elastic–plastic (Schubert, Thornton), elastic–dissipative (Sadd), plastic–dissipative (Walton) and viscoplastic–adhesion (Rumpf) contacts were also shown. With respect to these theories, a general ap-proach for the time- and deformation-rate-dependent and combined viscoelastic, plastic, viscoplastic, adhesion and dissipative behaviors of a spherical particle contact was derived and explained.

As the main result, the adhesion force FH is found to be a function of the force contribution FH0 without any deformation plus a pre-consolidation or load-history-dependent term with the normal force FN. These linear and non-linear ap-proaches can be interpreted as general constitutive models of the adhesion force. It should be pointed out here that the adhesion force level discussed in this paper is approximately 105–106 times the particle weight of nanoparticles. This means, in other words, that one has to apply these large values as acceleration ratio a/g with respect to gravity to separate these pre-consolidated contacts or to remove mechanically such adhered particles from solid surfaces.

For colliding particles a correlation between particle impact velocity and con-tact displacement response is obtained using energy balance. These constitutive model approaches are generally applicable for micro- and nanocontacts of particu-late solids [91, 148, 149]. Hence, these contact models are intended to be applied for modern data evaluation of product quality characteristics such as powder flow properties, i.e., yield loci, consolidation and compression functions or design of characteristic processing apparatus dimensions [122, 142–151].


Acknowledgements

The author would like to acknowledge his coworkers Dr. S. Aman, Dr. T. Gröger, Dr. W. Hintz, Dr. Th. Kollmann and Dr. B. Reichmann for providing relevant in-formation and theoretical tips. The advices from Prof. H.-J. Butt and Prof. S. Lud-ing with respect to the fundamentals of particle and powder mechanics were espe-cially appreciated during the collaboration of the project “shear dynamics of cohesive, fine-disperse particle systems” of the joint research program “Behavior of Granular Media” of the German Research Association (DFG).

Symbol Unit Description a nm contact separation A nm2 particle contact area aF=0 nm minimum center separation for molecular force equilib-

rium Ca – capillary number CH J Hamaker constant [69] based on interacting molecule pair

additivity CH,sls J Hamaker constant according to Lifshitz theory [70] for

solid–liquid–solid interaction d µm particle diameter or particle size (in powder technology) E kN/mm2 modulus of elasticity F N force FH nN adhesion force FH0 nN adhesion force of a rigid contact without any deformation FN nN normal force FT nN tangential force G kN/mm2 shear modulus h mm zone height hK nm height of overlap, indentation or center approach k – coordination number kN N/mm contact stiffness in normal direction kT N/mm contact stiffness in tangential direction m kg mass p kPa contact pressure pf MPa plastic micro-yield strength of particle contact pVdW MPa attractive van der Waals pressure r µm particle radius rK Nm contact radius t h time v m/s particle velocity

J. Tomas 42

vH m/s critical sticking velocity vR m/s bounce velocity W J deformation work Wm J/g mass related energy absorption by inelastic deformation δ nm tangential contact displacement ε – porosity ηK Pa⋅s particle contact viscosity κ – elastic–plastic contact consolidation coefficient, see

Eq. (71) κp – plastic repulsion coefficient, see Eq. (43) κp,t – viscoplastic repulsion coefficient, see Eq. (61) κvis – total viscoplastic contact consolidation coefficient, see

Eq. (74) µi – coefficient of internal friction, i.e., Coulomb friction ν – Poisson ratio ϕ i deg angle of internal friction between particles ρ kg/m3 density σ kPa normal stress σM kPa center stress of Mohr circle [1, 149] σR kPa radius stress of Mohr circle [1, 149] σsls mJ/m² surface tension of solid–liquid–solid interaction σt kPa tensile stress σ0 kPa isostatic tensile strength of the unconsolidated powder σ1 kPa major principal stress σ2 kPa minor principal stress τ kPa shear stress Φ T – dimensionless bond strength according to Tabor [74] ψ – loading parameter according to Thornton [53]

Indices A detachment- or contact-area-related at attraction b bulk br brittle c compressive crit critical diss dissipation e effective el elastic


f flow or yield F=0 potential force equilibrium (potential minimum) H adhesion i internal iso isostatic K particle contact l liquid m mass related M center min minimum N normal p pressure related pl plastic r micro-roughness R radius rep repulsion s solid S surface, shear sls solid-liquid-solid interaction between particles ss solid-vacuum-solid interaction between particles st stationary Sz shear zone t loading time dependent T tangential th theoretical tot total U unloading V volume related VdW van der Waals vis total viscoplastic 0 initial, zero point (0) beginning of iterations 1,2 particle 1, particle 2 3 mass basis of cumulative distribution of particle diameter (d3) 50 median particle diameter, i.e., 50% of cumulative distribution

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J. Tomas 44

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