Simulation of sticking of cohesive particles under normal impact

Simulation of sticking of cohesive particles under normal impact Raimondas Jasevičius1,a ,Rimantas Kačanauskas1,b, Jürgen Tomas2,c 1

Vilnius Gediminas Technical University, Lithuania, Magdeburg University, Germany E-mail: [email protected], [email protected], [email protected] 2Otto-von-Guericke

Abstract. Sticking of cohesive spherical particles under normal impact is investigated numerically by applying the Discrete Element Method. The nonlinear-dissipative contact model with adhesion is applied to model normal contact forces. Loading is described by elastic and elastic-plastic Hertz contact model. Damping is described by nonlinear Tsuji model. Adhesion is of non-linear character. Sticking and detachment behaviours for various damping values are considered in details. Influence of adhesion force for wide range of particle size is illustrated by variation of critical sticking velocity. Comparison for purely elastic behaviour is also presented.

Key words: cohesive granular material, normal elastic-plastic contact, sticking and detachment, critical sticking velocity, discrete element method

Interaction of particles described by the Hertz contact theory is usually used to describe repulsive contact forces independently on specific particle size. Fundamentals of the DEM and particular models of non-cohesive granular material may be found in [4-7] while important details of simulation technique and software implementation in [8-12]. Investigation of the nature of cohesive particles required to apply new models. Independently on development of the discrete element method, an expression of the normal force for spherical particles has been suggested by Johnson, Kendall and Roberts [13]. This JKR model assumes that the surface attraction force only results in a change of surface energy over the contact area. Derjaguin, Muller and Toporov et al. [14] considered a DMP model where surface attraction forces have a finite range and, therefore, act just outside the contact zone where surface separation is small. First comprehensive physical models for fine particles were published by Molerus [15, 16]. The earliest discussions about simplest adhesion (surface attraction) models and the importance of these effects to the results of the simulation of granular material behaviour are presented by Thornton and Yin [17] and Kohring [18]. Comprehensive research on cohesive powders is continued by Tomas [19-21]. Fundamentals of cohesive powder consolidation and flow are generalised in [22]. Here, the model of "stiff particles with soft contacts" was used for fine to ultra-fine particles. This hysteretic model includes the elastic-plastic particle contact behaviours with adhesion and load-unload. Some other interesting contribution and

Introduction The handling of granular materials is attached of great importance in pharmaceutical, food, cement, chemical and other industries. The problems relevant to particle segregation, the effects of granular material vibration, attrition, breakage, dust explosions and other phenomena have been encountered during the operation period of many technological devices. The granular state of the material is a transient state between the gas, liquid or solid [1-2], when the material possesses some of their properties and behaves similarly or in a completely different manner. Historically, major part of granular materials is treated as assembly of non-cohesive grains. The rapid increasing production of poly-dispersed dry cohesive powders make technical problems much more serious like undesired adhesion in particle conversion or powder handling, and desired in agglomeration or coating. Thus, understanding the fundamentals of particle adhesion with respect to product quality assessment and process performance is very essential in powder technology. Cohesive granular materials are currently being studied by applying experimental, theoretical and numerical methods. Recently, the discrete (distinct) element method (DEM) introduced by Cundall and Strack [3] has become a powerful tool for solving many scientific and engineering powder technology problems. It started with its first application to simulate the dynamic behaviour of non-cohesive granular material, which is presented as an assembly of grains.

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Presence of the load attraction force providing aggregation ability is characteristic feature of dry cohesive particulates. Concept of the impact of cohesive particles is explained by considering nonlinear elastic-plastic-dissipative contact behaviour. Our focus is on the model of isotropic, stiff spherical particles that are approaching to soft contacts. Thus, contact displacement (overlap size) h is assumed to be small compared to the size (diameter) d of the stiff particle. The evaluation of the inter-particle forces addresses evaluation of separate force Fij acting between particles i and j. Hence, the interaction force Fij found between two particles i and j is expressed in terms of the normal component Fij ≡ Fijn .

DEM simulations may be outlined. Shear behaviour of cohesive powders with friction was studied by Luding [23]. Cohesive powders combining contact elasticity and distant van der Waals-type attraction were simulated by Gilabert et al. [24]. Collision dynamics of granular particles with viscoelastic adhesion are considered by Brilliantov et al. [25] focusing basically on characterization of both restitutive collisions described by the coefficient of restitution as well as aggregative collisions described by the critical aggregative impact velocity. Elastic-plastic adhesion of ultrafine powders were considered by Tomas [26-27]. The paper addresses simulation of sticking of cohesive particles under normal impact. The non- elastic and elastic-plastic Hertz contact models with non-linear adhesion and Tsuji damping is applied to model normal contact forces. Sticking and detachment behaviours are illustrated in details. Influence of adhesion force for wide range of particle size is illustrated by variation of critical sticking velocity.

Constitutive model comprising force-displacement relations in form of algebraic functions has to be elucidated for these purposes. Typically, load-displacement diagram is plotted in the range of nanoscale. It has the first loading path and series of hysteric loops. Hysteric path comprises a series of unloading-reloading cycles. The applicability of this model is based on the quasistatic assumption which implies a relative impact rate much smaller than the material’s speed of sound. This allows us to treat the collision process as a sequence of equilibrium states. Conceptually, the normal contact may be characterized by combining classical visco-elastic "spring-dashpot" model [3] supplied by adhesion (Fig. 1a). The spring model is assumed to be history independent and accumulation effect are not considered. Here, pressure and compression are defined as positive but tension and extension are negative. Consequently, a normal interaction force during binary collision of particles comprises three components of slightly different nature

Simulation methodology The DEM methodology based on the Langrangian approach is applied to simulate dynamic behaviour of the cohesive particles under normal impact. From modelling point of view, spherical particles considered are termed hereafter as discrete elements. When moving, the particles impact and deform each other. The motion of arbitrary particle i is characterized by a small number of global parameters: positions xi , velocities

x& i = dxi dt and accelerations &x&i = d 2 xi dt 2 of the mass center and force applied to it. Translational motion is described by the Newton's second law applied to each particle i

m i &x& i (t ) = Fi (t )

n n n Fijn = Fspring + Fadh + Fdiss

(2)

(1) n where contact deformation, or spring, force Fspring , adhesion

where mi is the mass, while vector Fi presents the resultant force act on the particle i. It may comprise prescribed and contact forces. Rotational motion, if necessary may be described in the same manner. Methodology of calculating of the contact forces in (1) depends on the particle geometry and mechanical properties as well as on the constitutive model of the particle interaction. Detailed description of models employed here will be presented below. The integration of differential equations (1) for particle i at the time t+Δt (where Δt is the time step) is performed numerically by applying the 5th order Gear’s predictor-corrector scheme [4, 6, 28].

n n force Fadh and dissipation, or dashpot, force Fdiss . Various linear and non-linear models may be applied to evaluate particular force components. Realistic and theoretically motivated nonlinear contact is governed by the Hertz contact theory. According to which the elastic repulsion force n Fspring = Feln of the contacting particles depends on the

overlap length by the power of α = 3/2 and can be expressed as:

Feln = K eln h α

Normal contact

(3)

Here the nonlinear stiffness constant is

The mechanical behaviour of cohesive particles is essentially different from non-cohesive granular material.

K eln =

2

4 eff eff 2−α Eij (Rij ) 3

(4)

non-cohesive interaction. Generally, dissipation is based on analogy with that hold in the Hertz theory. Thus, non-liner damping force reads n n & Fdiss = C diss h

(6)

Various approaches exist to evaluate the nonlinear n characteristic Cdiss , witch is displacement depending damping coefficient and h& is displacement rate, respectively. Kuwabara and Kono [30] intuitively proposed a fully nonlinear model combining spring and dissipative forces. This model was also independently derived by Brilliantov et al. [31]. Tsuji et al. [32] proposed a Hertz-type force law including a slightly modified dissipative term with a different n exponent. Finally, non-linear dissipative constant Cdiss is expressed in terms of reduced mass and stiffness (4):

a)

n n Cdiss = α d mijeff K spring h1 / 4

where αd is adjustable non – dimensional damping coefficient. Adhesive contact starts, when distance between particles is h ≥ a F 0 , which means the long-range adhesion force without any contact deformation (the so-called jump in). This model presents the simplest case of a realistic van der Waals force. Particle approach curve contains two essential parameters: maximum attractive force Fh 0 and minimum

b)

separation range a F 0 . Particles i and j still attract each other if the gap h between their surfaces is smaller than the separation range 0 ≥ h ≥ a F 0 . When considering the smooth sphere–sphere contact without any contact deformation particle approach is formed by attractive adhesion force

Fig. 1. Models of normal contact of adhesive spherical particles: a) elastic model, b) elastic plastic model

Here Eijeff =

Ei E j

Ei (1 − ν ) + E j (1 − ν 2 i

2 j

)

;

(7)

Rijeff =

Ri R j Ri + R j

;

(5) n Fadh =

present the effective contact elasticity modulus and the effective radius of particles i and j, while Ei and Ej are particular elasticity moduli and νi and νj are Poisson’s ratios, respectively. Unloading, beginning at arbitrary point U’, is assumed to recover deformation path defined by (3-5). In presence of arbitrary dissipation mechanism, unloading-reloading will follow different path U’-A. In order to reflect energy dissipation, the normal force n may directly contain the viscous dissipation term Fdiss , as shown in (2). Review and systematic analysis of known and the new extended models for normal contact and comparison to available experimental data are presented by KruggelEmden et al. [29]. Here, different dissipation mechanisms and their applicability are discussed, restricting, however, to

Fh 0 a F2 0 (a F 0 − h )2

(8)

As follows from the Fig. 1a, at detachment point A′ with overlap hA , contact unloading curve crossed adhesion limit curve. This curve reflects elastically - plastically deformed, annular contact zone [26, 27]. Finally, it is represented by formula: n Fadh = _ lim

2 πRijeff p f κ p hA Fh 0 a F 0 3 − aF 0 2 3 (a F 0 + hA − h ) (a F 0 + hA − h )

(9)

Here, a second term expresses influence of accumulated plastic deformation during contact. The dimensionless plastic repulsion coefficient κ p describes a dimensionless ratio of

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attractive van der Waals pressure of a plate–plate model to a constant repulsive micro-yield strength p f .

mobile particle is initially subjected by the portion of induced energy which is controlled by initial impact velocity υ0. Impact velocity serves the base for initial conditions. During impact characterised by loading path Fh 0 -U’ (Fig. 1b) particles collide and undergo elastic and later elastic-plastic deformation while induced kinetic energy is transformed to elastic deformation energy and partially dissipated. At certain time instant, the mobile particle reaches the state of the rest characterized by zero velocity υ = 0 and the maximal overlap h = hmax. After reaching the maximal overlap, mobile particle starts to separate and follows unloading path U’-A’. Behaviour of cohesive particles during unloading largely depends on the imposed energy. If energy is high, at certain time instant elastic and dissipative forces exceed n adhesive limit F n < Fadh and the particles begin to detach _ lim

The model for the normal contact under consideration is based on the resent developments of Tomas [26, 27]. It combines nonlinear elastic – plastic contact behaviour and load dependent non-linear model of adhesion. The new features of this model are explained in a previous manner. Referring Fig. 1b it is obvious, that the contact may be initially loaded from point Fh 0 to Y and, as the response, is elastically deformed with an approximated circular contact area. In elastic stage, nonlinear contact behaviour is governed by Hertzian model (3-4). With increasing external normal load the soft contact starts at a pressure pf with plastic yielding at the point Y. Particles overlap hY at point Y can be calculated by Tomas [26, 27] formula:

 3πp f (κ A − κ p ) hY = Rijeff   2 Eijeff  

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(Fig. 2c). For smaller impact rates a different scenario is observed. If imposed energy is insufficient to exceed adhesive potential, particles remain stick together (Fig. 2d) and compression-tension oscillates, while kinetic energy is dissipated by viscous deformations until equilibrium id reached.

(10)

Here, κ A is dimensionless elastic – plastic contact area coefficient representing the ratio of plastic particle contact deformation area to total contact deformation area and includes a certain plastic displacement. As a consequence, the spring force in loading is defined by generalised expression n Fspring = βel Feln + βel − pl Feln− pl

(11)

Contact coefficients βel and βel− pl depends on contact specification. When contact elastic βel = 1 and βel − pl = 0 ,

Fig. 2. Behaviour of particle during impact: a) loading b) unloading , c) detachment, d) sticking

when contact is elastic – plastic then coefficients exchanges. Finally, elastic – plastic contact force reads

Feln− pl = πRijeff p f (κ A − κ p )h

Particular parameters reflecting microscopic properties of cohesive particles are: adhesion force of the spheresphere contact Fh0 = -2.64 nN, micro yield strength pf = 300 MPa, plastic repulsion coefficient κp = 0.153, elastic-plastic contact area coefficient κA = 5/6. Fife values of damping factor αd1 = 0; αd2 = 0.2; αd3 = 0.3; αd4 = 0.4; αd5 = 0.5 of the Tsuji model were explored in simulations. Basing on results of Tsuji et al. [31], damping factor values applied were chosen in order to reflect identical coefficients of restitution due to viscous dissipation equal to ξ1 = 1.0; ξ2 ≈ 0.76; ξ3 ≈ 0.65; ξ4 ≈ 0.56; ξ5 ≈ 0.48. The case of zero damping αd = 0; illustrates non-viscous behaviour. Influence of impact energy was investigated by considering two values of the impact velocity: υ01 = 2 mm/min and υ02 = 2 m/min. Illustration of contact behaviour in terms of load-displacement relationship for various damping ratios is given in Fig. 3. Here, normal force comprises all interparticle (right hand) forces, including viscous dissipation.

(12)

It is obvious that yield limit can not be exceeded; however, at certain point U unloading can begin. The nonlinear spring stiffness are extracted from (11-12). Constitutive expressions (3-12) serve the base for simulation of impact. Numerical investigation of normal contact Normal contact during impact of two identical spherical particles is considered numerically. Numerical experiment assumes mobile particle impacting the fixed target particle. Contact behaviour is considered by integrating equations of motion (1) and applying various combinations of elasticplastic-dissipative contact models with adhesion (3-12). Numerical experiment conducted with particles defined by constant radius Rijeff = 0.3 μm is illustrated in Fig. 2. The

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The higher impact velocity is of restitutive character and results detachment of particles occurring after the single loading–unloading loop (Fig. 3a). In opposite, the low velocity is of aggregative character and leads to sticking and following highly hysteric oscillations (Fig.3b). Present approach deals with numerical DEM simulations supported by sensitivity analysis. Two types of contact models for range of effective particle radius Rijeff varying between 0.1 and 1 μm for both elastic and elasticplastic models combined with dissipative Tsuji damping model (7-8) were examined. The elastic loading path was defined by (3-6), while elastic-plastic loading path was defined by (10-11). Finally, variation critical sticking velocity as function of particle radii in logarithmic scale is illustrated in Fig. 4. a)

Fig. 4: Variation of critical sticking velocity versus effective radius for different loading models

b)

Here, the straight lines 1’, 2’, 3’ and 4’ exhibit results of elastic model, while curves 1, 2, 3 and 4 illustrate increased role of elastic-plastic deformation for smaller particles. Line numbers indicate the values of damping factor αd2 = 0.2; αd3 = 0.3; αd4 = 0.4; αd5 = 0.5. The character of graphs meet the tendency reported in [25]. Concluding remarks Characteristic feature of normal collision of cohesive particles exhibiting aggregative and restitutive behaviour was studied numerically by applying DEM. Influence of adhesion for wide range of effective particle radius varying between 0.1 and 1 μm was examined by considering both elastic and elastic-plastic contact models and illustrated by variation of critical sticking velocity. On the basis of numerical results some conclusions could be drawn. Generally, adhesion significantly affects aggregation of smaller particles. The critical sticking

c) Fig. 3. Illustration of contact behaviour of particles during normal impact: a) υ0 = 2 mm/min; b) υ0 = 2 m/min, c) sticking behaviour when αd = 0.2;

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[12] Balevičius, R., Markauskas, D. Numerical stress analysis of granular material. – Mechanika, v. 4, No 66, p. 12-17, 2007. [13] Johnson K. L., Kendall K., Roberts A. D. Surface energy and the contact of elastic solids. Proc. R. Soc. Lond. A, p. 301-313, 1971. [14] Derjuguin B. V., Muller V. M., Toporov Y. P. Effect of contact deformations on the adhesion of particles. Journal of Colloid and Interface Science, 53(2), p. 314- 325, 1975. [15] Molerus O. Theory of yield of cohesive powders. Powder Technol., 12, p. 259-275, 1975. [16] Molerus O. Effect of interparticle cohesive forces on the ow behaviour of powders. Powder Technol., 20, 161-175, 1978. [17] Thornton C. and Yin K. K. Impact of elastic spheres with and without adhesion. Powder Technol., 65, p. 153-166, 1991. [18] Kohring G.A. Computer simulations of granular materials: the effects of miesoscopic forces. Phys. I France, 4, 17791782, 1994. [19-20] Tomas J. Assessment of mechanical properties of cohesive particulate solids: Part 1. Particle contact constitutive model. Part. Sci. Technol., 19, 95- 110, 2001. Part 2. Powder flow criteria, Part. Sci. Technol., 19, p. 111- 129, 2001. [21] Luding S., Tykhoniuk R., Tomas J. Anisotropic material behaviour in dense, cohesive powders. Chem. Eng. Technol., 26(12), p. 1229- 1232, 2003. [22] Tomas J. Fundamentals of cohesive powder consolidation and flow. Granular Matter 6, p. 75-86, 2004. [23] Luding S. Shear flow modeling of cohesive and frictional fine powder. Powder Technology, 158(1-3), 45-50, 2005. [24] Gilabert F. A., Roux J.-N., Castellanos A. Computer simulation of model cohesive powders: Inuence of assembling procedure and contact laws on low consolidation states. Physical review E, 75, 011303, p. 1-25, 2007. [25] Brilliantov V.N., Albers N., Spahn F., Poschel T. Collision dynamics of granular particles with adhesion. Physical Review E, 76, 051302, 2007. [26] Tomas J. Adhesion of ultrafine particles. A micromechanical approach. Chemical Engineering Science, 62, p. 1997- 2010, 2007. [27] Tomas J. Adhesion of ultrafine particles. Energy absorption at contact. Chemical Engineering Science, 62, p. 5925- 5939, 2007. [28] Jasevičius R., Kačianauskas R. Modelling deformable boundary by spherical particle for normal contact. Mechanika, 6(68). p. 513, 2007. [29] Kruggel-Emden H., Simsek E., Rickelt S., Wirtz S., Scherer V. Review and extension of normal force models for the Discrete Element Method. Powder Technol., 171(3), p. 157-173, 2007. [30] Kuwabara G., Kono K. Restitution coefficient in a collision between two spheres. Japanese Journal of Applied Physics, 26, p. 1230-1233, 1987. [31] Brilliantov N. V., Spahn F., Hertzsch J., Poeschel T. Model for collisions in granular gases. Physical Review E, 53(5), p. 5382-5392, 1996. [32] Tsuji Y., Tanaka T., Ishida T. Lagrangian numerical simulation of plug of cohesionless particles in a horizontal pipe. Powder Technol., 71, p.239-250, 1992.

velocity in the range particles effective radius varying between 0.1 and 1 μm. Critical sticking velocity υ cs decreases if particle effective radius is higher and for millimeter range particles, where critical sticking velocity and adhesion effects have no account. Development of plastic deformation observed in soft contact model increases critical sticking velocity and considerably deviates from elastic solution by decreasing of particle size. Increase of viscous damping even more accelerates this tendency. Assumptions on non-linear adhesion simplified spring model neglects, however, additional dissipation mechanisms, and future research is still required. Acknowledgements This work originated during visit of the first author in Magdeburg supported by German Academic Exchange Servise under Grant ref. No. 323, PKZ/A 0692650. References [1] Jaeger H.M., Nagel S.R., Behringer R.P. Granular solids, liquids, and gases. Reviews of Modern Physics 68 (4) p. 1259-1273, 1996. [2] Herrmann H.J. Granular matter, Physica A 313. p. 188 – 210, 2002. [3] Cundall P.A. and Strack O.D.L. A discrete numerical model for granular assemblies. Geotechnique, 29(1), p. 47-65, 1979. [4] Allen M.P. and Tildesley. D.J. Computer simulation of liquids. Oxford Science Publication, 1987. [5] Herrmann H.J. and Luding S. Modelling granular media on the computer. Continuum Mech. Thermodyn, 10, p. 189-231, 1998. [6] Džiugys A. and Peters B. J. An Approach to Simulate the Motion of Spherical and Non-Spherical Fuel Particles in Combustion Chambers. Granular Material, 3(4), p. 231-266, 2001. [7] Pöschel T. and Schwager T. Computational granular dynamics, Models and algorithms. Springer, Berlin, 2004. [8] Balevičius R., Džiugys A., Kačianauskas R. Discrete element method and its application to the analysis of penetration into granular media, Journal of Civil Engineering and Management 10 (1) p. 3-14, 2004. [9] Balevičius R., Kačianauskas R., Džiugys A., Maknickas A., Vislavičius V. DEMMAT code for numerical simulation of multi-particle dynamics. Information Technology and Control, 34(1) p. 71-78, 2005. [10] Balevičius R., Džiugys A., Kačianauskas R., Maknickas A., Vislavičius K. Investigation of performance of programming approaches and languages used for numerical simulation of granular material by the discrete element method. Computer Physics Communications 175, p. 404-415, 2006. [11] Maknickas A., Kačeniauskas A. Kačianauskas R., Balevičius R., Džiugys A. Parallel DEM Software for Simulation of Granular Media. Informatica, 17(2), p. 207-224, 2006.

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