Particle-particle interactions and models (Discrete Element Method) Stefan Luding MSM, TS, CTW, UTwente, NL Granular Materials Real: sand, soil, rock, grain, rice, lentils, powder, pills, granulate, micro- and nano-particles Model Granular Materials steel/aluminum spheres spheres with dissipation/friction/adhesion 1
Classification Particle size range Classified as 0.1-1.0 µm Ultra-fine powder 1.0 10 µm Superfine powder 10-100 µm Granular powder 0.1 3.0 mm Granular solid 3.0 10 mm Broken solid Powders Granular materials Usual working range Richards and Brown (1970), Nedderman (1992) Particle-particle interaction Point Contact Pore Structure Relative Movement Electrical Conductivity Thermal Conductivity Permeability Shearing Motion Fluidisation 2
2D stress Stress distribution at contact Stress distribution over the contact area between two spheres pressed together under a normal force N 3
Powder and Liquid Flow (differences) Inherent Yield Stress Powders heap Liquid spreads Yield stress = resistance against flow Particle Interactions Mechanical (d p >10µm) Chemical (10nm<d p <10µm) Atomic Cluster (d p <10nm) 4
Surface and Field Forces Particle Interactions Material Connections Mechanical (d p >10µm) Chemical (10nm<d p <10µm) Atomic Cluster (d p <10nm) Two-phase Flows 5
Two phase Flows How to approach? Experiments Continuum theory (materials, micropolar, ) Statistical Physics + Kinetic theory + dissipation + friction Numerical Modeling Monte Carlo (stochastic methods) Molecular dynamics-like simulations (DEM) Continuum like FEM, CFD, SPH, 6
Deterministic or Stochastic Models? Method Molecular dynamics (soft particles) Event Driven (hard particles) Monte Carlo (random motion) Direct Simulation Monte Carlo Lattice (Boltzmann) Models Abbrev. MD ED MC DSMC LB Theory (Kinetic Theory) Stat. Phys. Kinetic Theory Navier Stokes Deterministic or Stochastic Models? Method MD (soft p.) Determ./ Stochast. D Discrete Time X Discrete Space Discrete Events Flexible ***** Fast * ED (hard p.) D X * *** MC S? * ** DSMC S X *** **** LB S X X * ***** 7
Flow in porous media Deterministic Models Method Molecular dynamics (soft particles) Event Driven (hard particles) Monte Carlo (random motion) Direct Simulation Monte Carlo Lattice (Boltzmann) Models Abbrev. MD ED MC DSMC LB Theory (Kinetic Theory) Stat. Phys. Kinetic Theory Navier Stokes 8
Approach philosophy Introduction Single particle Single Particles Particle Contacts/Interactions Contacts Many particle cooperative behavior Applications/Examples Conclusion Many particle simulation Continuum Theory What is DEM? 1. Specify interactions between bodies 2. Compute all forces f j i 3. Integrate the equations of motion for all particles mɺɺ x = f i j i j i 9
What is DEM? 1. Specify interactions between bodies (for example: two spherical atoms) 2. Compute all forces f j i 3. Integrate the equations of motion for all particles (Verlet, Runge-Kutta, Predictor-Corrector, ) with fixed time-step dt mɺɺ x = f i j i j i Linear Contact model f - (really too) simple - linear - very easy to implement hys i k1δ for un-/re-loading = 10
Inclined plane Hopper Flow 11
Convection Segregation 12
Segregation Mixing Reverse segregation P. V. Quinn, D. Hong, SL, PRL 2001 Linear Contact model f - (really too) simple - linear - very easy to implement hys i k1δ for un-/re-loading = 13
Hertz Contact model - simple limited rel. - non-linear - easy to implement f hys i k δ = 3/ 2 1 for un-/re-loading Anisotropy 3D 3D 14
Sound 3-dimensional modeling of sound propagation P-wave shape and speed Normalized adhesion force (mn/m) Contact force measurement (PIA) 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 Normalized load (mn/m) 15
Contact Force Measurement Adhäsions [nn] Adhesion and Friction Adhesion force (nn( nn) 85 80 75 70 65 20 30 40 50 60 70 Relative Luftfeuchte rel. humidity (%) rel. humidity (%) Friction force (nn( nn) ) N (n ft ra sk u n g e ib R 500 400 300 200 100 0-1000-750-500 -250 0 250 500 750 Andruckkraft (nn) Normal force (nn( nn) 16
Hysteresis (plastic deformation) Deformation Hysteresis in nm 20 15 10 5 0 Polystryrene -5 0 50 100 150 200 250 Force in nn Deformation Hysteresis in nm 20 15 10 5 0 Borosilicate glass -5 0 1000 2000 3000 Force in nn Collaborations: MPI-Polymer Polymer Science (Butt et al.) Contact properties via AFM Contacts 1. loading 17
Contacts 1. loading plastic loading stiffness: k 1 Contacts 1. loading plastic loading stiffness: k 2. unloading elastic un/re-loading stiffness: k 1 * 18
Contacts 1. loading plastic loading stiffness: k 2. unloading 3. re-loading elastic un/re-loading stiffness: k 1 * Contacts 1. loading plastic loading stiffness: k 2. unloading 3. re-loading elastic un/re-loading stiffness: k 4. tensile failure tensile force 1 * 19
Contacts 1. loading plastic loading stiffness: k 2. unloading 3. re-loading elastic un/re-loading stiffness: k 4. tensile failure tensile force 1 * Contacts 1. loading transition to stiffness: k 2. unloading 3. re-loading elastic un/re-loading stiffness: k 4. tensile failure max. tensile force 2 2 20
Adhesion forces Two phase Flows with Aggregation 21
Tangential contact model Sliding contact points: - Static Coulomb friction - Dynamic Coulomb friction v t t ( v ) ˆ i v j n ( aiωi + a jω j ) a ˆ ijn ( ωi ω j ) a ˆ ˆ ijnn ( ωi ω j ) + = sliding rolling torsion Tangential contact model - Static friction - Dynamic friction - spring - dashpot ( ) project into tangential plane ϑ = ϑ nˆ nˆ ϑ compute test force f = k ϑ γ ɺ ϑ and tˆ = f 0 0 0 t t t t t f sticking: sliding: f f µ f 0 t s n 0 t > µ s d fn f f t t = f µ 0 t = ˆt d fn ( t t ) kt ϑ = ϑ + ɺ ϑ dt ϑ = f + γ ɺ ϑ before contact static dynamic static 22
Tangential contact model Rolling (mimic roughness or steady contact necks) - Static rolling resistance - Dynamic resistance v t t ( v ) ˆ i v j n ( aiωi + a jω j ) a ˆ ijn ( ωi ω j ) a ˆ ˆ ijnn ( ωi ω j ) + = sliding rolling torsion Tangential contact model Torsion (large contact area) - Static torsion resistance - Dynamic resistance v t t ( v ) ˆ i v j n ( aiωi + a jω j ) a ˆ ijn ( ωi ω j ) a ˆ ˆ ijnn ( ωi ω j ) + = sliding rolling torsion 23
Silo Flow Initial Outflow Jamming Silo Flow with friction µ = 0.5 µ = 0.5 µ = 0.2 r 24
Silo Flow with friction µ = 0.5 µ = 0.2 r Details of interaction Attraction + Dissipation = Agglomeration 25
Literature (http://www2.msm.ctw.utwente.nl/sluding/publications.html) [1] S. Luding Collisions & Contacts between two particles, in: Physics of dry granular Media, eds. H. J. Herrmann, J.-P. Hovi, and S. Luding, Kluwer Academic Publishers, Dordrecht, 1998 [http://www2.msm.ctw.utwente.nl/sluding/papers/coll2p.pdf] [2] S. Luding, Introduction to Discrete Element Methods: Basics of Contact Force Models and how to perform the Micro-Macro Transition to Continuum Theory, European Journal of Environmental and Civil Engineering - EJECE 12 - No. 7-8 (Special Issue: Alert Course, Aussois), 785-826 (2008), [http://www2.msm.ctw.utwente.nl/sluding/papers/luding_alert2008.pdf] [3] S. Luding, Cohesive frictional powders: Contact models for tension Granular Matter 10(4), 235-246, 2008 [http://www2.msm.ctw.utwente.nl/sluding/papers/ludingc5.pdf] [4] M. Lätzel, S. Luding, and H. J. Herrmann, Macroscopic material properties from quasi-static, microscopic simulations of a twodimensional shear-cell, Granular Matter 2(3), 123-135, 2000 [http://www2.msm.ctw.utwente.nl/sluding/papers/micmac.pdf] [5] S. Luding, Anisotropy in cohesive, frictional granular media J. Phys.: Condens. Matter 17, S2623-S2640, 2005 [http://www2.msm.ctw.utwente.nl/sluding/papers/jpcm1.pdf] Discrete element model (DEM) Equations of motion 2 d ri mi = f 2 i dt Forces f and = f torques: + f + m g c w i c i w i i Contacts Contact if Overlap > 0 1 δ = d + d r r n Overlap ( i j ) ( i j ) 2 ɵ ( ri rj ) Normal n = n = ij r r i j Many particle simulation 26
f i f = mɺɺ δ = kδ + γδɺ i ij overlap ( i j ) ( i j ) 2 rel. velocity ɺ δ = ( vi v j ) n δ 1 δ = d + d r r n δ = a a n acceleration ɺɺ ( i j ) Linear Contact model - really simple - linear, analytical - very easy to implement http://www2.msm.ctw.utwente.nl/sluding/papers/coll2p.pdf f i f = mɺɺ δ = kδ + γδɺ i ij overlap ( i j ) ( i j ) 2 rel. velocity ɺ δ = ( vi v j ) n δ 1 δ = d + d r r n acceleration ɺɺ δ ( ) Linear Contact model - really simple - linear, analytical - very easy to implement ( f j = f i ) f f j 1 = a i i a j n = n = fi n mi m j mij http://www2.msm.ctw.utwente.nl/sluding/papers/coll2p.pdf 27
f = mɺɺ δ = kδ + γδɺ i ij kδ + γδɺ + mɺɺ ij δ = 0 k m ij γ δ + 2 ɺ δ + ɺɺ δ = 0 2m 0 + 2 ɺ + ɺɺ = 0 2 ω δ ηδ δ ij Linear Contact model - really simple - linear, analytical - very easy to implement ω = elastic freq. 0 k mij eigen-freq. visc. diss. 2 2 ω = ω0 η γ η = m 2 ij http://www2.msm.ctw.utwente.nl/sluding/papers/coll2p.pdf f = mɺɺ δ = kδ + γδɺ i ij kδ + γδɺ + mɺɺ ij δ = 0 k m ij γ δ + 2 ɺ δ + ɺɺ δ = 0 2m 0 + 2 ɺ + ɺɺ = 0 2 ω δ ηδ δ ω = elastic freq. 0 eigen-freq. visc. diss. ij k mij 2 2 ω = ω0 η γ η = m 2 ij Linear Contact model - really simple - linear, analytical - very easy to implement v 0 δ ( t) = exp( ηt )sin( ωt) ω ɺ v0 δ ( t) = exp( ηt )[ η sin( ωt) ω + ω cos( ωt) ] contact duration t π c = ω v( tc ) restitution coefficient r = v = exp( 0 ηt ) http://www2.msm.ctw.utwente.nl/sluding/papers/coll2p.pdf c 28
Linear Contact model (m w = ) ω = elastic freq. 0 eigen-freq. visc. diss. particle-particle k mij 2 2 ω = ω0 η γ η = 2m ij particle-wall wall k ω ω 0 0 = = m wall 2 2 ω = ω0 2 η 4 wall γ η η = = 2 2 m i i 2 contact duration tc π ω = restitution coeff. r = exp( ηt c ) t wall c = π wall > t ω wall wall wall r = exp( η t c ) c http://www2.msm.ctw.utwente.nl/sluding/papers/coll2p.pdf Time-scales time-step contact duration t <= tc t c 50 π ω = t n < t c t wall c = π wall > t ω c time between contacts t n > t sound propagation N t... with number of layers N L c c L experiment T http://www2.msm.ctw.utwente.nl/sluding/papers/coll2p.pdf 29
Time-scales time-step contact duration t <= tc t c 50 n c = l arg e small tc > tc π ω t < t time between contacts different sized particles t n > t sound propagation N t... with number of layers N L c c L experiment T http://www2.msm.ctw.utwente.nl/sluding/papers/coll2p.pdf Force-chains experiments - simulations 30
Force-chains experiments - simulations Bi-axial box (stress chains) 31
Granular flow - compressibility? / dilatancy? - MohrCoulomb-like yield stress? - shear viscosity? - inhomogeneity? (shear-band) - (almost always) an-isotropy? - micro-polar effects (rotations) Bi-axial box (kinetic energy) 32
Granular flow - compressibility? / dilatancy? - MohrCoulomb-like yield stress? - shear viscosity? - inhomogeneity? (force-chains) - (almost always) an-isotropy? - micro-polar effects (rotations) Bi-axial box (rotations) 33
3D - Minimal Density? Variation with friction ν=0.48 Saturation at large friction coefficients 3D - Minimal Density? Variation with rolling resistance ν=0.45 Saturation at large rolling resistance 34
3D - Initial Condition Density? van der Waals adhesion ν=0.20-0.400.40 clusters/agglomerates form 3D - Initial Condition Density? van der Waals adhesion ν=0.20-0.400.40 clusters/agglomerates form 35
Details of interaction Attraction + Dissipation = Agglomeration T(t) Sintering (pressure and temperature) 1. Preparation 2. Heating 3. Sintering 4. Cooling 5. Relaxation 6. Testing 36
Preparation p=20000 p=2000 p=200 p=20 p=2 Sintering cold contacts 1. Preparation (T=const.) k2max k2 (δ ) loading stiffness: k1 = k1 (T ) contact stiffness: k2 = k2 (δ ) 37
Sintering 2 2. Heating 1 1 ( ) loading stiffness: k = k T maximum overlap fixed: δ + max neutral overlap increasin g: δ + 0 Sintering 2 2. Heating 1 1 ( ) loading stiffness: k = k T maximum overlap fixed: δ + max neutral overlap increasin g: δ + 0 38
Sintering 3 3. Sintering Sintering 3 3. Sintering t 0 = 10s - slow dynamics (t 0 ) - diffusion, - trick: increase t 0 time delay: ( ) (, ) ( ) 2 k1 T k1 T t k1 ( T, t) = ± t k1 T t0 1 t k1 ( T, t) = k1 ( T ) 1 1 k1 ( T0 ) k1 ( T ) t 0 1 39
Sintering 4 4. Cooling Sintering 4 4. Cooling maximum overlap increasing : δ neutral overlap fixed: δ 0 max 40
Sintering 5 5. Relaxation Contact forces after Sintering after Relaxation 41
Sintering 6 6. Testing strain p=const. Sintering 6 6. Testing strain p=const. 42
Sintering 6 6. Testing strain p=const. Sintering 6 6. Testing strain p=const. 43
Sintering 6 6. Testing strain p=const. cracks Sintering 6 Density Shrinkage! 44
Sintering 6 Stiffness sintering time p=100 p=10 Sintering 6 6. Contact network p=100 p=10 45
Sintering 7 7. Vibration test p=100 p=10 PCT (pressure-compression-tension) 1. Preparation 2. HIGH pressure 3. Relaxation 4. Compression 5. Tension p(t) p s 46
uni-axial compression-tension Compression Tension compression - uni-axial k k 2 = 1 2 t 47
compression - uni-axial k k 2 = 1 2 t compression - uni-axial k k 2 = 1 2 t 48
compression - uni-axial k k 2 = 1 2 t tension - uni-axial k k 2 = 1 2 t 49
tension adhesion stiffness: kt k2 k k 2 = 1 5 k k 2 = 1 2 t t k k = k k 2 = 20 t 2 1 t uni-axial tension Compression Tension 50
healing (compression) 1. Preparation 2. HIGH pressure 3. Relaxation 4. Compression 5. Tension 6. Healing Olaf Herbst, PostDoc healing (tension) 1. Preparation 2. HIGH pressure 3. Relaxation 4. Compression 5. Tension 6. Healing Olaf Herbst, PostDoc 51
healing (tension) 1. Preparation 2. HIGH pressure 3. Relaxation 4. Compression 5. Tension 6. Healing Olaf Herbst, PostDoc Another example: Membranes Add strings to the hard spheres 52
Another example: Membranes Add strings to the hard spheres Another example: Membranes Add strings to the hard spheres P 53
Membranes gravity shear -flow Cylinder: Weak Electric Field z x y E E x y z 54
x Cylinder: Strong Electric Field E z y x y z 55