Mechanics of nanoparticle adhesion A continuum approach

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1 Particles on Surfaces 8: Detection, Adhesion and Removal, pp Ed..L. Mittal VSP 003 Mechanics of nanoparticle adhesion A continuum approach JÜRGEN TOMAS Mechanical Process Engineering, Department of Process Engineering and Systems Engineering, Otto-von-Guericke-University, Universitätsplatz, D Magdeburg, Germany Abstract The fundamentals of particle-particle adhesion are presented using continuum mechanics approaches. The models for elastic (Hertz, Huber, Cattaneo, Mindlin and Deresiewicz), elasticadhesion (Derjagin, Bradley, Johnson), plastic-adhesion (rupp, Molerus, Johnson, Maugis and Pollock) contact deformation response of a single, normal or tangential loaded, isotropic, smooth contact 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 deformation 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 deformation and a plate plate model for this micro-contact flattening is presented. Various contact deformation 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 socalled contact pre-consolidation history. Finally, for colliding particles the correlation between particle impact velocity and contact deformation response is obtained using energy balance. This constitutive model approach is generally applicable for solid micro- or nanocontacts but has been shown here for dry titania nanoparticles. eywords: Powder; particle mechanics; contact behavior; constitutive models; adhesion force; nanoparticles. 1. INTRODUCTION In terms of particle processing and product handling, the well-known flow problems 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) , Fax: (49-391) , juergen.tomas@vst.uni-magdeburg.de

2 J. Tomas feeding and dosing, as well as undesired effects such as widely spread residence time distribution, time consolidation or caking, chemical conversions and deterioration of bioparticles. Finally, insufficient apparatus and system reliability of powder processing plants are also related to these flow problems. The rapid increasing production of cohesive to very cohesive nanopowders, e.g., very adhering pigment particles, micro-carriers in biotechnology or medicine, auxiliary materials 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, ]), the yield locus concept of Jenike [3, 4] and Schwedes [5], the Warren Spring equations [6 10], and the approach by Tüzün [1], 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 [0] 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 [1], Campbell [], Walton [3, 4], Herrmann [5] and Thornton [6]. Additionally, particle adhesion effects are related to undesired powder blocking at conveyer transfer chutes or in pneumatic pipe bends [7] in powder handling and transportation, to desired particle cake formation on filter media [8, 9], to wear effects of adhering solid surfaces [30, 31], fouling in membrane filtration, fine particle deposition in lungs, formulation of particulate products [3 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 approach 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.. 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):

3 Mechanics of nanoparticle adhesion A continuum approach 3 (1) elastic contact deformation (Hertz [41], Huber [4], Derjaguin [43], Bradley [44, 45], Cattaneo [46], Mindlin [47], Sperling [48], rupp [49], Greenwood [50], Johnson [51], Dahneke [5], Thornton [53, 54], Sadd [55]), which is reversible, independent of deformation rate and consolidation time effects and valid for all particulate solids; () plastic contact deformation with adhesion (Derjaguin [43], rupp [56], Schubert [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], rupp [49], Rumpf et al. [6] 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. [6]), which is irreversible and dependent on deformation rate and consolidation time, e.g., nanoparticles fusion. 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 (h /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 describe the force-displacement behavior in terms of nanomechanics. The microscopic particle shape remains invariant during the dynamic stressing and contact deformation at this nanoscale. In powder processing, these particles are manufactured 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 industrial practice..1. Elastic, plastic and viscoplastic contact deformations.1.1. Elastic contact displacement For a single elastic contact of two spheres 1 and with a maximum contact circle radius r,el but small compared with the particle diameter d 1 or d, an elliptic pressure distribution p el (r ) is assumed, Hertz [41]: pel = 1 r (1) pmax r, el With the maximum pressure p(r = 0) = p max in the center of contact circle at depth z = 0, p 3 F =, N max π rel ()

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9 Mechanics of nanoparticle adhesion A continuum approach 9 and the median particle radius r 1, (characteristic radius of contact surface curvature) (Fig. 1), r 1, 1 1 = + r1 r and the average material stiffness (E, modulus of elasticity; ν, Poisson ratio) 1 (3) * E 1 ν 1 = + E 1 ν 1 E one can calculate the correlation between normal force F N and maximum contact radius r,el : 3 r F E 3 1, N = el, * r Considering surface displacement out of the contact zone (for details, see Huber [4]) the so-called particle center approach or height of overlap of both particles h is [41]:, el 1, 1 (4) (5) 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): * 3 FN = E r 3 1, h (7) Eq. (7) is shown as the dashed curve marked Hertz. The maximum pressure p max, Eq. (), is 1.5 times the average pressure F N /( π r, el) on the contact area and lies below the micro-yield strength p f. Because of surface bending and, consequently, the opportunity for unconfined yield at the surface perimeter outside of the contact circle r r,el (Fig. 1b), a maximum tensile stress is found [4] σ tmax, 015. pmax (here negative because of positive pressures in powder mechanics): t 1 ν r max 3 r r rel, σ p, el = (8) This critical stress for cracking of a brittle particle material with low tensile strength is smaller than the maximum shear stress τ max 031. pmax according to Eq. (11) which is found at the top of a virtual stressing cone below the contact

10 10 J. Tomas Figure 1. Characteristic spherical particle contact deformation. (a) Approach and (b) elastic contact deformation (titania, primary particles d = nm, surface diameter d S = 00 nm, median particle diameter d 50,3 = 610 nm, specific surface area A S,m = 1 m²/g, solid density ρ s = 3870 kg/m 3, surface moisture X W = 0.4%, temperature θ = 0 C) [148]. Pressure and compression are defined as positive but tension and extension are negative. The origin of this diagram (h = 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 = a 0 = a F=0. After approaching from an infinite distance to this minimum separation a F=0 the sphere sphere contact without any contact deformation is formed by the attractive adhesion force F H0 (the so-called jump in ). Then the contact may be loaded F H0 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 according to Huber [4] at the perimeter of contact circle is neglected for the elliptic pressure distribution, drawn below panel b.

11 Mechanics of nanoparticle adhesion A continuum approach 11 area on the principal axis r = x + y = 0 in the depth of z r,el /. Combining the major principal stress distribution, Eq. (9), σ 1 = σ z (z) at contact radius r = 0 max rel + el, σ z ( z ), = p r z and the minor principal stress, Eq. (10), σ = σ y = σ t (z) [4], el, (9) t ( z) 1 rel, rel, = + ( 1+ ν ) 1 z arctan max r r rel, z, el σ p r z, (10) + the maximum shear stress inside a particle contact r r,el is obtained using the τ = σ σ [1]: Tresca hypothesis for plastic failure ( ), el max 1 ( z) 3 rel, 1 ν rel, = + 1 z arctan max 4 r r rel, z, el τ p r z + (11) This internal shear stress distribution becomes more and more critical for ductile or soft solids with a small transition to yield point, and consequently, plastic contact deformation like nanoparticles with very low stiffness, see Section.3. Due to the parabolic curvature F N (h ), the particle contact becomes stiffer with increasing diameter r 1,, contact radius r or displacement h (k N is the contact stiffness in normal direction): df k = = E r h = E r (1) N * * N dh 1, 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 contact force as [53, 54]: ( ) * 4 ψ δ 1 ψ tanϕ T 1, i N F = G r h ± F (13) Here δ is the tangential contact displacement, ψ the loading parameter dependent on loading, unloading and reloading, ϕ i the angle of internal friction, G= E 1+ ν the shear modulus, and the averaged shear modulus is given as: ( ) * ν1 ν G = + G1 G 1 (14)

12 1 J. Tomas Thus, with ψ = 1 the ratio of the initial tangential stiffness dft * kt = = 4 G r dδ (15) to the initial normal stiffness according to Eq. (1) is: k k T N ( ν ) 1 = ν Hence this ratio ranges from unity, for ν = 0, to /3, for ν = 0.5 [63], which is different from the common linear elastic behavior of a cylindrical rod..1.. 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 F H,DMT / 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, endall and Roberts (JR) model [67], which assumes that all the interactions occur within the contact radius of the particles. The median adhesion force F H,DMT (index H,DMT) of a direct spherical contact can be expressed in terms of the work of adhesion W A, conventional surface energy γ A or surface tension σ sls as WA = γ A = σ sls. The index sls means particle surface-adsorption layers (with liquid equivalent mechanical behavior) particle surface interaction. If only molecular interactions with separations near the contact contribute to the adhesion force then the so-called Derjaguin approximation [43] is valid F HDMT, sls 1, (16) = 4 π σ r, (17) which corresponds to Bradley s formula [44]. This surface tension σ sls equals half the energy needed to separate two flat surfaces from an equilibrium contact distance a F=0 to infinity [75]: σ 1 CHsls, sls = pvdw ( a) da= a 4 π af = 0 (18) F = 0 The adhesion force per unit planar surface area or attractive pressure p VdW which is used here to describe the van der Waals interactions at contact is equivalent to a theoretical bond strength and can simply calculated as [75] (e.g., p VdW MPa): p C 4 σ Hsls, sls VdW = = 3 6 π a a F = 0 F = 0 (19)

13 Mechanics of nanoparticle adhesion A continuum approach 13 Using this and for a comparison of the adhesion or bond strength, a dimensionless ratio of adhesion displacement (extension at contact detachment), expressed as height of neck h N,T around the contact zone, to minimum molecular center separation a F=0 can be defined as h NT, 4 r1, σ sls ΦT = = a F = 0 * 3 E a F = 0 1/ 3, (0) 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 particles Φ T < 0.1 [66, 76, 8]. For separating a stiff, non-deformed spherical point contact, the DMT theory [65] predicts a necessary pull-off force F N,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 σ sls = mj/m or the Hamaker constant according to the Lifshitz continuum theory amounts to C H,sls = (0. 40) 10 0 J [70, 75]. Notice here, the particle interactions depend greatly on the applied load, which is experimentally confirmed by atomic force microscopy [78, 79, 98]. A balance of stored elastic energy, mechanical potential energy and surface energy delivers the contact radius of the two spheres [51], expressed here with a constant adhesion force F H,JR from Eq. (3): (,,, ) 3 r r F F F F F E 3 1, = * N HJR HJR N HJR (1) Eq. (1) indicates a contact radius enhancement with increasing work of adhesion. The contact force-displacement relation is obtained from Eqs. (6) and (1) and can be compared with the Hertz relation Eq. (7) by the curves in Fig. marked with Hertz and JR: * * E 3 4 E FH,JR 3 N 1, 1, F = r h r h () 3 3 For small contact deformation the so-called JR limit [51] is half of the constant adhesion force F H,JR. This JR model can be applied for higher bond strengths Φ T > 5 [76, 8 84]. It is valid for comparatively larger and softer particles than the DMT model predicts [115]: FHJR, FNZJR,, = = 3 π σ sls r1, (3) Thus the contact radius for zero load F N = 0,

14 14 J. Tomas Figure. 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 p max = p f at the yield point Y then the contact starts with plastic yielding which is intensified by mobile 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 A el (thickness r,el ) and circular plastic A pl (radius r,pl ) contact area, shown below panel c.

15 Mechanics of nanoparticle adhesion A continuum approach 15 3 r 3 1, FHJR, 0, * r = (4) E is reduced to the pull-off contact radius, i.e., 3, pull off =, 0 4 r - r /. (5) Additionally, with applying an increasing tangential force F T, the contact radius r is reduced by the last term within the square root: r F F F F F E 4 G * 3 3 r1, FT E = N+ HJR, + * HJR, N+ HJR, *, (6) When the square root in Eq. (6) disappears to zero, a critical value F T,crit is obtained, the so-called peeling of contact surfaces [64, 88]: F ( HJR, N HJR, ) F F + F G = (7) E Tcrit, * An effective or net normal force (F N + F H,JR ) remains additionally in the contact [54]. Considering F T > F T,crit, i.e., contact failure by sliding (see Mindlin [63]), the tangential force limit is expressed as FT = tanϕi ( FN + FH0). The adhesion force F H0 (index H0) is constant during contact failure and the coefficient (or angle) of internal friction µ i = tanϕ i is also assumed to be constant for a multi-asperity contact [50, 81, 8]. This constant friction was often confirmed for rough surfaces in both elastic and plastic regimes [50, 81, 84], but not for a single-asperity contact with nonlinear dependence of friction force on normal load [8, 84]. Rearranging Eq. (6), the extended contact force displacement relation shows a reduction of the Hertz (first square-root) and JR contributions to normal load F N which is needed to obtain a given displacement h : * * * E 3 4 E FHJR, 3 FT E N = 1, 1, * F r h r h G * (8) 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 adhesion 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-

16 16 J. Tomas 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 and 3. If the maximum pressure p max = p f in the center of the contact circle reaches the micro-yield strength, the contact starts with irreversible plastic yielding (index f). From Eqs. () and (5) the transition radius r,f and from Eq. (6) the center approach h,f are calculated as: h π r p = (9) E 1, f f, * r π r1, pf f, * = (30) E Figure 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. [6] and Molerus [13, 14] introduced this philosophy in powder mechanics and the JR theory was the basis of adhesion mechanics [58, 67, 76, 85, 86, 90] 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 ( ) 1 1 1, 1 1, 1, 1 1, r = r r h = r h h d h (31) the total particle center approach of the two spheres: r r r h = h1, + h, = + = (3) d d r 1 1, Because of this, a linear force displacement relation is found for small spherical particle contacts. The repulsive force as a resistance against plastic deformation is given as: F = p A = π d p h (33) Npl, f 1, f Thus, the contact stiffness is constant for perfect plastic yielding behavior, but decreases with smaller particle diameter d 1, especially for cohesive fine powders and nanoparticles: df k = = d p (34) N π Npl, dh 1, f

17 Mechanics of nanoparticle adhesion A continuum approach 17 Figure 3. Characteristic particle contact deformation. (d) Elastic unloading and reloading with dissipation (titania) [148]. After unloading U E the contact recovers elastically in the compression mode and remains with a perfect plastic displacement h,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 h,a. The reloading curve runs from point A to U to the displacement h,u, Eq. (65).

18 18 J. Tomas Additionally, the rate-dependent, perfect viscoplastic deformation (at the point of yielding) expressed by contact viscosity η times indentation rate h is assumed to be equivalent to yield strength p f multiplied by indentation height increment h p h = η h (35) f and one obtains again a linear model regarding strain rate: F = η A = π d η h (36) Nvis, 1, An attractive viscous force is observed, e.g., for capillary numbers Ca = η h / σ > 1 when comparatively strong bonds of (low-viscous) liquid lg bridges are extended with negative velocity h [71 73]. Consequently, the particle material parameters: contact micro-yield strength p f and viscosity η are measures of irreversible particle contact stiffness or softness. Both plastic and viscous contact yield effects were intensified by mobile adsorption layers on the surfaces. The sum of deformation increments results in the energy dissipation. For larger particle contact areas A, the conventional linear elastic 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?.. Particle contact consolidation by varying adhesion force rupp [49] and Sperling [48, 56] developed a model for the increase of adhesion force F H (index H) of the contact. This considerable effect is called here as consolidation and is expressed as the sum of adhesion force F H0 according to Eq. (17) plus an attractive/repulsive force contribution due to irreversible plastic flattening of the spheres (p f is the repulsive microhardness or micro-yield strength of the softer contact material of the two particles, σ ss /a 0 is the attractive contact pressure, index ss represents solid vacuum solid interaction): σ FH = 4 π r 1, σss 1+ a p ss 0 f (37) Dahneke [5] modified this adhesion model by the van der Waals force without any contact deformation F H0 plus an attractive van der Waals pressure (force per unit surface) p VdW contribution due to partially increasing flattening of the spheres which form a circular contact area A (C H is the Hamaker constant based on interacting molecule pair additivity [69, 75]):

19 Mechanics of nanoparticle adhesion A continuum approach 19 H 1, H = H0 + VdW = a a 0 0 F F A p C r h (38) The distance a 0 denotes a characteristic adhesion separation. If stiff molecular interactions are provided (no compression of electron sheath), this separation a 0 was assumed to be constant during contact loading. By addition the elastic repulsion of the solid material according to Hertz, Eq. (7), to this attraction force, Eq. (38), and by deriving the total force F tot with respect to h, the maximum adhesion force was obtained as absolute value F CH r1, CH r1, Hmax, = 1+ 6 a * E a0 which occurs at the center approach of the spheres [5]: h CH r1, max, = * 6 9 E a0, (39) (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 F N [57]. Generally, if this external compressive normal force F N 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. c, and the adhesion force between these two partners increases, see in Fig. 3 the so-called adhesion boundary for incipient contact detachment. During this surface stressing the rigid particle 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. [6] have developed a constitutive model approach to describe the linear increase of adhesion force F H, mainly for plastic contact deformation: ( ) pvdw pvdw FH = 1+ FH0 + FN = 1+ κp FH0 + κ p F (41) N pf pf With analogous prerequisites and derivation, Molerus [14] obtained an equivalent expression:

20 0 J. Tomas p F = F + F = F + F (4) VdW κ H H0 p N H0 p N f The adhesion force F H0 without additional consolidation (F N = 0) can be approached 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 F N to a plate plate contact (Fig. c). The coefficient κ p describes a dimensionless ratio of attractive van der Waals pressure p VdW for a plate plate model, Eq. (19), to repulsive particle micro-hardness p f which is temperature sensitive: p C κ (43) VdW H,sls p = = p 3 f 6 π af = 0 pf This is referred to here as a plastic repulsion coefficient. The Hamaker constant C H,sls for solid liquid solid interaction (index sls) according to Lifshitz theory [70] is related to continuous media which depends on their permittivities (dielectric 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 = a F=0 ) or interaction potential minimum [75, 76, 91]. Its magnitude is about a F= nm. This separation 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 a F=0 is assumed to be constant during loading and unloading for technologically relevant powder pressures σ < 100 kpa (Fig. c). For a very hard contact this plastic repulsion coefficient is infinitely small, i.e., κ p 0, and for a soft contact κ p 1. If the contact circle radius r is small compared to the particle diameter d, the elastic and plastic contact displacements can be combined and expressed with the annular elastic A el and circular plastic A pl contact area ratio [57]: p F F F VdW H = H0 + N A el pf 1+ 3 A pl (44) For a perfect plastic contact displacement A el 0 and one obtains again Eq. (4): F F + κ F (45) H H0 p N This linear enhancement of adhesion force F H with increasing preconsolidation force F N, Eqs. (41), (4) and (45), was experimentally confirmed for micrometer sized particles, e.g., by Schütz [94, 95] (κ p = 0.3 for limestone) and Newton [96] (κ p = for poly(ethylene glycol), κ p = for starch, κ p =

21 Mechanics of nanoparticle adhesion A continuum approach for lactose, κ p = for CaCO 3 ) with centrifuge tests [9] as well as by Singh et al. [97] (κ p = 0.1 for poly(methylmethacrylate), κ p 0 for very hard sapphire, α-al O 3 ) with an Atomic Force Microscope (AFM). The two methods are compared with rigid and rough glass spheres (d = µm), without any contact deformation, by Hoffmann et al. [98]. Additionally, using the isostatic tensile strength σ 0 determined by powder shear tests [91, 1, 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 F H 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, df N,Z,br /df N (F N,Z,br F H ), for a comparatively small surface energy σ ss of the rigid sphere gold plate contact (index ss). In contrast, a pull-off force F N,Z,br proportional to F N was obtained from the JR theory [58] for the full plastic range of high loading and brittle separation of the contact (Table 1): F F * N N,Z,br = σss E 3 π pf (46) Additionally, a load-dependent adhesion force was also experimentally confirmed 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 particlesurface contacts and, consequently, asperity stressing by simultaneous normal pressure and shearing, contact deformation, microcrack initiation and propagation, and microfracture of brittle silicon asperity peaks affect directly the polishing performance. Thus the Coulomb friction becomes dominant also in a wet environment..3. Variation in adhesion due to non-elastic contact consolidation.3.1. Elastic plastic force displacement model All interparticle forces can be expressed in terms of a single potential function Fi =± Ui( hi)/ hiand thus are superposed. This is valid only for a conservative system in which the work done by the force F i versus distance h i is not dissipated as heat, but remains in the form of mechanical energy, simply in terms of irreversible deformation, e.g., initiation of nanoscale distortions, dislocations or lattice 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 U / h = 0 one obtains the potential-force balance. ij ij i j

22 J. Tomas Thus, the elastic plastic force displacement models introduced by Schubert et al. [57], Eq. (44), and Thornton [60] Eq. (47) ( ) F N =π p f r 1, h h,f / 3 (47) should be supplemented here with a complete attractive force contribution due to contact flattening described before. Taking into account Eqs. (41), (4) and (44), the particle contact force equilibrium between attraction (-) and elastic plus, simultaneously, plastic repulsion (+) is given by ( r * represents the coordinate of annular elastic contact area): H0 VdW π N f π,pl F = 0 = F p r F + p r r,pl * * * el + π p ( r ) r dr r 3/ (48) π pmax r r,pl FN + F H0 + pvdw π r = pf π r,pl (49) r At the yield point r = r,pl the maximum contact pressure reaches the yield strength p el = p f. π r pf FN + F H0 + pvdw π r = pf π r,pl + 3 (50) pmax Because of plastic yielding, a pressure higher than p f is absolutely not possible and thus, the fictitious contact pressure p max is eliminated by Eq. (1): π r r,pl FN + FH0 + p VdW π r = pf π r,pl + 1 (51) 3 r Finally, the contact force equilibrium r,pl + + = π + 1 N H0 VdW f 3 3 r F F p A p r and the total contact area A are obtained: A = p A + 1 f 3 3 A pl (5)

23 Mechanics of nanoparticle adhesion A continuum approach 3 A = FN + FH0 A 1 pl pf p A VdW (53) Next, the elastic plastic contact area coefficient κ A is introduced. This dimensionless coefficient represents the ratio of plastic particle contact deformation area A pl to total contact deformation area A = Apl + A and includes a certain elastic el displacement: κ 1 pl A = + A 3 3 A (54) The solely elastic contact deformation A pl = 0, κ A = /3, has only minor relevance for cohesive powders in loading (Fig. ), but for the complete plastic contact deformation (A pl = A ) the coefficient κ A = 1 is obtained. From Eqs. (43), (53) and (54) the sum of contact normal forces is obtained as: ( ) N + H0 = f A p F F π r p κ κ (55) From Eq. (5) the transition radius of elastic-plastic model r,f,el-pl (index el-pl) and from Eq. (6) the particle center approach of the two particles h,f,el-pl are calculated as: h ( ) 3 π r p κ κ 1, f A p =,f,el pl * r E ( ) 9 π r p κ κ 1, f A p,f,el = pl * 4 E (56) (57) Checking this model, Eq. (56), with pure elastic contact deformation, i.e., κ p 0 and κ A = /3, the elastic transition radius r,f, Eq. (9), is also obtained. For example, nanodisperse titania particles (d 50,3 = 610 nm is the median diameter on mass basis (index 3), E = 50 kn/mm modulus of elasticity, ν = 0.8 Poisson ratio, p f = 400 N/mm micro-yield strength, κ A 5/6 contact area ratio, κ p = 0.44 plastic repulsion coefficient) a contact radius of r,f,el-pl =.1 nm and, from Eq. (57), a homeopathic center approach of only h,f,el-pl = 0.03 nm are obtained. This is a very small indentation calculated, in principle, by means of a continuum approach. The contact deformation is equivalent to a microscopic force F N =.1 nn or to a small macroscopic pressure level of about σ = 1.4 kpa (porosity ε = 0.8) in powder handling and processing.

24 4 J. Tomas 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 κ A constant: ( ) F + F = π r p κ κ h (58) N H0 1, f A p But if one considers the contact area ratio of Eq. (63), a slightly nonlinear (progressively increasing) curve is obtained. Using the elastic plastic contact consolidation coefficient κ due to definition (Eq. (71)) one can also write: π r p κ F F h 1+ κ 1, f A N + H0 = The curve of this model is shown in Fig. 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 h,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 viscoplastic contact, one obtains deformation with Eqs. (36) and (58) (κ A constant): F + F = π r η κ κ h (60) ( ) N H0 1, A p,t (59) A dimensionless viscoplastic contact repulsion coefficient κ p,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 yielding. κ p,t p = η h VdW The consequences for the variation in adhesion force are discussed in Section.3.3 [147]..3.. 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 displacement, h,e, obtained in combination with Eq. (58): 3 =,E,U,f,U (61) h h h h (6)

25 Mechanics of nanoparticle adhesion A continuum approach 5 Thus, the contact area ratio κ A is expressed more in detail with Eqs. (6) and (54) for elastic κ A = /3 and perfect plastic contact deformation, κ A = 1 if h,u : h h κ,e,f = + = A 3 3 h,u 3 h (63),U Beyond point E to point A, the same curve runs down to the intersection with the adhesion boundary, Eq. (67), to the displacement h,a : ( ) 3 * FN,unload = E r 3 1, h h,a F H,A (64) Consequently, the reloading runs along the symmetric curve ( ) 3 * FN,reload = E r 3 1, h,u h + F N,U (65) from point A to point U to the displacement h,u as well (Fig. 3). The displacement h,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 h,u : ( κ ) h 3,A,(1) = h,u h,f,el- pl h,u + h,a,(0) (66) If one replaces F N in Eq. (7) (see Section.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, h = h,a compared with Eq. (38) and point A in the diagram of Fig. 4: F = F + π r p h (67) H,A H0 1, VdW,A The unloading and reloading hysteresis for an adhesion contact takes place between 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 a= af = 0 + h,a h. The actual particle separation a can be used by a long-range hyperbolic adhesion force curve F a 3 N,Z with the van der Waals pressure p VdW as given in Eq. (19) and the displacement h,a for incipient contact detachment by Eq. (66):

26 6 J. Tomas Figure 4. Characteristic particle contact deformation. (e) Contact detachment [148]. Again, if one applies a certain pull-off force F N,Z = F H,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 a= af = 0 + h, A h. This actual particle separation is considered for the calculation by a 3 hyperbolic adhesion force curve F N,Z = F a of the plate plate model Eq. (68). HA,

27 Mechanics of nanoparticle adhesion A continuum approach 7 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, preconsolidation level. Thus, the variation in adhesion forces between particles depends directly on this frozen irreversible deformation, the so-called contact pre-consolidation history. F F + π r p h H0 1, VdW,A N,Z( h ) = 3 h,a h 1 + a F= 0 af= 0 (68) These generalized functions in Fig. 3 for the combination of elastic-plastic, adhesion and dissipative force displacement behaviors of a spherical particle contact were derived on the basis of the theories of rupp [49], Molerus [13], Maugis

28 8 J. Tomas [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, ] as the history-dependent contact behavior 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.1.3). Thus, the adhesion force increases with interaction time [3, 49, 77, 18]. This time-dependent consolidation behavior (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 [1 15, ]. This method refers more to incipient sintering or contact fusion of a thermally-sensitive particle material [6] without interstitial adsorption layers. This micro-process is very temperature sensitive [1, 14, 15, 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 repulsion κ p + κ p,t, line 5 in Table. 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. Material parameters for characteristic adhesion force functions F H (F N ) in Fig. 8 Constitutive model of contact deformation Repulsion coefficient Constitutive models of combined contact deformation Contact area ratio Contact consolidation coefficient Intersection with F N -axis (abscissa) Instantaneous contact consolidation plastic κ p C VdW Hsls, p = p = 3 f 6 π a F = 0 p f elastic plastic A pl Time-dependent consolidation viscoplastic κ pt, pvdw = t η elastic plastic and viscoplastic κ A = + κ 3 At, = + 3 ( Apl + A 3 el ) 3 ( Apl + Avis + Ael ) κ p κ = κ κ A p F a h p NZ, π F= 0 r 1, f f ( CHsls, ) κ F vis A pl κp + κp, t = κ κ κ NZtot,, + A At, p pt, ( CHsls, ) vis π a h p 1+ p t / η f F= 0 r1, f f

29 Mechanics of nanoparticle adhesion A continuum approach 9 Figure 6. Characteristic elastic plastic, viscoelastic viscoplastic particle contact deformations (titania, primary particles d = nm, surface diameter d S = 00 nm, median particle diameter d 50,3 = 610 nm, specific surface area A S,m = 1 m /g, solid density ρ s = 3870 kg/m 3, surface moisture X W = 0.4%, temperature θ = 0 C, loading time t = 4 h). The material data, modulus of elasticity E = 50 kn/mm, modulus of relaxation E = 5 kn/mm, relaxation time t relax = 4 h, plastic microyield strength p f = 400 N/mm, contact viscosity η = Pa s, Poisson ratio ν = 0.8, Hamaker constant C H,sls = J, equilibrium separation for dipole interaction a F=0 = nm, contact area ratio κ A = 5/6 are assumed as appropriate for the characteristic contact properties. The plastic repulsion coefficient κ p = 0.44 and viscoplastic repulsion coefficient κ p,t = 0.09 are recalculated from shear-test data in a powder continuum [147, 149].

30 30 J. Tomas Figure 7. Constitutive models of contact deformation of smooth spherical particles in normal direction without (only compression +) and with adhesion (tension ). The basic models for elastic behavior 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 expanded stepwise to include nonlinear plastic contact hardening and softening equivalent to shearthickening and shear-thinning in suspension rheology [91]. Energy dissipation was considered by Sadd et al. [55] and time-dependent viscoplasticity by Rumpf et al. [6]. Considering all these theories, one obtains a general contact model for time- and rate-dependent viscoelastic, plastic, viscoplastic, adhesion and dissipative behaviors [91, , 151].

31 dentation t < / ( p ) Mechanics of nanoparticle adhesion A continuum approach 31 η κ, e.g., t < 5 d for the titania used as a very cohesive f powder (specific surface area A S,m = 1 m /g, with a certain water adsorption capacity). All the material parameters are collected in Table. A viscoelastic relaxation in the particle contact may be added as a timedependent function of the average modulus of elasticity E *, Yang [61] and rupp [49] (t relax is the characteristic relaxation time): 1 = exp * * * * E E ( ) ( 0) t E0 t = E ( tt ) 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 distributed material functions. Usually one may focus here only on the characteristic or averaged values of these constitutive functions 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, 1, 149]. Replacing the contact area in Eq. (38), the following force force relation is directly obtained: relax p F + F VdW H0 N FH = FH0 + pvdw A = FH0 + p 1 A p A f pl VdW A pf p (69) (70) Therefore, with a so-called elastic plastic contact consolidation coefficient κ, κ p κ = (71) κ κ a linear model for the adhesion force F H as function of normal force F N is obtained (Fig. 8): F κ F κ F F F ( 1 κ ) A p H = H0 + N = + H0 + κ κ N A κp κa κp (7) The dimensionless strain characteristic κ is given by the slope of adhesion force F H which is influenced by predominant plastic contact failure. It is a meas-

32 3 J. Tomas Figure 8. Particle contact forces for titania powder (median particle diameter d 50,3 = 610 nm, specific surface area A S,m = 1 m /g, surface moisture X W = 0.4%, temperature = 0 C) according to the linear model Eq. (7), non-linear model Eq. (79) for instantaneous consolidation t = 0 and the linear model for time consolidation t = 4 h (Eq. (73)). The powder surface moisture X W = 0.4% is accurately analyzed with arl Fischer titration. This is equivalent to an idealized mono-molecular adsorption layer being in equilibrium with an ambient air temperature of 0 C and 50% humidity. ure of irreversible particle contact stiffness or softness. A shallow slope designates a low adhesion level F H F H0 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 displacement rate (Section.1.3). Thus, the contact reacts softer and, consequently, the adhesion level is higher than before. This new adhesion force slope κ vis is modified by the viscoplastic contact repulsion coefficient κ p,t, which includes certain viscoplastic microflow at the contact (Table and Fig. 8), κ κ + κ F = F + F At, p p, t Htot, κ κ κ H0 κ κ κ N At, p pt, At, p pt, ( 1 κ ) = + F + κ F vis H0 vis N (73) with the so-called total viscoplastic contact consolidation coefficient κ vis that includes the elastic-plastic κ p and the viscoplastic contributions κ p,t of contact flattening,

33 Mechanics of nanoparticle adhesion A continuum approach 33 κ vis κp + κp,t = κ κ κ A,t p p,t (74) F=0 r 1, Eqs. (7) and (73) consider also the flattening response of soft particle contacts at normal force F N = 0 caused by the adhesion force κ F H0 (rupp [56]) and κ vis F H0. Hence, the adhesion force F H0 represents the sphere sphere contact without any contact deformation at minimum particle surface separation a F=0. This initial adhesion force F H0 may also include a characteristic nanometer-sized height or radius of a rigid spherical asperity a < h << r with its center located at an average radius of the spherical particle, rupp [49], Rumpf et al. [33] and Schubert et al. [34]: F C h r / h C h H,sls r 1, r H,sls r H0 = af= 0 6 af= 0 1, 1, 1, ( 1+ hr / a ) 1, F= 0 This rigid adhesion force contribution F H0, see also the fundamentals [44, 69] and supplements [3, 49, 56, 67, 98 11], is valid only for perfect stiff contacts. Additionally, for mono-sized spheres d = 4 r an averaged asperity height 1, h r = h r can be used with ( ) 1 h = 1/ h + 1/ h. When the asperity height is 1, r1, r1 r not too far from the averaged sphere radius h < r, then the adhesion force can be calculated as, see Hoffmann (index Ho) [98]: F H0, Ho Hsls, r 6 a 1+ h / r F= 0 r 1, 1, ( ) 1, r1, 1, C h Hsls 1 r / h, r, 1 1, r1, = + 6 a 1+ h / r F = 0 r 1 1,, ( 1+ h / a ) r1 = 0, F C h 1, (75) (76) If the radius of the roughness exceeds the minimum separation of the sphere plate system in order of magnitudes (h r >> a F=0 ), the contribution of the plate, second term in Eq. (76), can be neglected and the adhesion force may be described as the sphere-sphere contact [98]. Rabinovich and co-workers [ ] have used the root mean square (RMS) roughness from AFM measurements and the average peak-to-peak distance between these asperities λ r to calculate the interaction between a smooth sphere and a surface with nanoscale roughness profile (index Ra):

34 34 J. Tomas F C r 1 1 H,sls 1, H0,Ra = + 6 a F= r1, RMS / λr ( RMS / af = 0 ) (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 nm is significant enough to reduce the theoretical adhesion force F H0 by an order of magnitude or more [115]. Greenwood [50, 80, 81] and Johnson [67] described the elastic and plastic deformations of random surface asperities of contacts by the standard deviation of roughness and mean pressure. The intersection of function (7) with abscissa (F H = 0) in the negative of consolidation force F N (Fig. 8), is surprisingly independent of the Hamaker constant C H,sls : F = π a h p κ 1+ NZ, F= 0 r1 f A, π a h p F= 0 r1, f r / h 1, r 1, ( 1+ h / a r ) 1, F= 0 (78) This minimum normal (tensile or pull-off) force limit F N,Z for nearly brittle contact failure combines the influences of the particle contact hardness p f (3 15) σ f (σ f = yield strength in tension, details in Ghadiri [117]) for a confined plastic micro-stress field in indentation [116] and the particle separation distribution, which is characterized here by the mean particle roughness height h, and the molecular center separation a F=0. Obviously, this value characterizes also the contact softness with respect to a small asperity height h r as well, see Eq. (34). This elastic plastic model (Eq. (7)) can be interpreted as a general linear constitutive contact model concerning loading pre-history-dependent particle adhesion, i.e., linear in forces and stresses, but non-linear regarding material characteristics. But if one eliminates the center approach h of the loading and unloading functions, Eqs. (58) and (64), an implied non-linear function between the contact pulloff force F H,A = F N,Z at the detachment point A is obtained for the normal force at the unloading point F N = F N,U : r 1,