Analysis of Cohesive Micro-Sized Particle Packing Structure Using History-Dependent Contact Models

Transcription

1 Analysis of Cohesive Micro-Sized Paricle Packing Srucure Using Hisory-Dependen Conac Models Raihan Tayeb, Xin Dou, Yijin Mao, Yuwen Zhang 1 Deparmen of Mechanical and Aerospace Engineering Universiy of Missouri Columbia, Missouri, 65211, USA Absrac Granular packing srucures of cohesive micro-sized paricles wih differen sizes and size disribuions, including mono-sized, uniform and Gaussian disribuion, are invesigaed by using wo differen hisory dependen conac models wih Discree Elemen Mehod (DEM). The simulaion is carried ou in he framework of LIGGGHTS which is a DEM simulaion package exended based on branch of granular package of widely used open-source code LAMMPS. Conac force caused by ranslaion and roaion, fricional and damping forces due o collision wih oher paricles or conainer boundaries, cohesive force, van der Waals force, and graviy are considered. The radial disribuion funcions (RDFs), force disribuions, porosiies, and coordinaion numbers under cohesive and non-cohesive condiions are repored. The resuls indicae ha paricle size and size disribuions have grea influences on he packing densiy for paricle packing under cohesive effec: paricles wih Gaussian disribuion have he lowes packing densiy, followed by he paricles wih uniform disribuion; he paricles wih mono-sized disribuion have he highes packing densiy. I is also found ha cohesive effec o he sysem does no significanly affec he coordinaion number ha mainly depends on he paricle size and size disribuion. Alhough he magniude of ne force disribuion is differen, he resuls for porosiy, coordinaion number and mean value of magniude of ne force do no vary significanly beween he wo conac models. Keywords: Granular Packing; Discree Elemen Mehod; Size disribuion; Radial Disribuion Funcion Nomenclaure d e diameer of paricle, m coefficien of resiuion g graviy, m/s 2 m v mass of paricle, kg velociy, m/s 1 Corresponding auhor. zhangyu@missouri.edu. 1

2 F force on paricle, N I momen of ineria, kg m 2 R T X Y radius of paricle, m orque, N m posiion vecor, m Young s modulus, Pa Greek Symbols γ θ μs μr ξn ξ damping coefficien, s roaional angle, rad sliding fricion coefficien rolling fricion coefficien normal direcion displacemen, m angenial displacemen, m ρ paricle densiy, kg/m 3 σ σp ω sandard deviaion Poisson raio angular velociy, rad/s Inroducion Granular packing simulaion is usually used o model srucures of maerials ha are involved in many indusrial applicaions ranging from manufacuring raw maerials o developing advanced producs. The impac of paricle properies on heir packing srucures is of he prime imporance o he enire packing process and is always essenial for fabricaion. A beer undersanding of packing is beneficial o opimize and o improve he indusrial applicaions. This opic has been inensively sudied in he pas decades; many of hem focused on he micro level packing [1-3] where packing densiy, which is equals o uniy minus is porosiy, is used as heir main indicaor o measure and evaluae qualiy of packing srucure [4]. Among hose works, researchers, by varying paricle sizes, size disribuions or forces involved in he packing process, obained deailed informaion of packing srucures and revealed weighed influences from differen parameers [5-9]. Some of hem are ineresed in cohesive effec ha is subsanially caused by cohesive forces such as van der Waals force, capillary force ha is associaed wih we paricles and elecrosaic force ha can be imporan for finer paricles. Cohesive effec urns ou o be of imporance in paricular siuaions, for example, when packing conainers are no longer rigid bu are kind of maerial ha has similar properies like dry sand, cemen or we soil, or i is no even 2

3 solid jus like seling paricles in he fluid where effec of graviy will reduce and cohesive effec will become significan [10-12]. The sudies on behaviors of cohesive paricles are usually carried ou by changing paricle sizes, mixure componen percenage if paricles are no made wih he same maerial or fluid densiy if paricles are seled down ino a fluid. Effecs of paricle diameers (mean diameers for mixed paricle cases) are considered o be a grea facor ha influences he packing srucures hereby worh more aenion. Boundary condiion is anoher imporan facor ha alers he force and deformaion of he packing srucure. Previously, researchers mosly adoped he periodic boundary condiions for he packing process where paricles ha exi he simulaion box will come back in opposie direcion in order o mainain he number of paricles in he simulaion box [13]. This approach allows ha he simulaion can be carried ou smoohly, because of lower chances of losing sysemic energy and generaing huge ineracive force. In his work, he cohesive effecs associaed wih size disribuions, which include mono-sized, uniform, and Gaussian disribuions, will be invesigaed by using wo differen hisory-dependen conac models. I is worh o poin ou ha he uniform disribuion of paricle size indicaes he sizes of paricles linearly increase from he minimum o he maximum. The range for uniform disribuion is kep a a consan of 40 μm. For Gaussian disribuion, he STD (sandard deviaion), σ, is se as μm for all cases. In addiion, fixed boundary condiions are applied o all sides of he simulaion box for all he cases such ha he paricles may collide wih boundaries during he packing process which will cause energy losses due o fricion. In order o undersand he fundamenals ha govern he cohesive paricle packing, a series of well-designed programs are developed based on he Discree Elemen Mehod [14-17]. LIGGGHTS[18] ha is based on LAMMPS [19], providing a simulaor of solving paricle relaed problems from indusrial applicaions, is employed o resolve he packing process. The simulaion resuls, including radial disribuion funcion (RDF) ha indicaes how number densiy changes wih disance from a seleced reference poin, force disribuions ha give a view on he magniude of forces acing on he paricles, porosiies and coordinaion numbers, are presened in his paper. Numerical Mehods and Physical Models I is well known ha any moions of a rigid paricle can be decomposed o wo pars: ranslaional and roaional moions. Referring o he Newon s second law, he governing equaions for each paricle during his packing process can be wrien as: 2 i m X i F (1) i 2 d θ Ii d 2 i 2 (2) T i F F F (3) n i ij ij T T T (4) r i ij ij where mi is he mass of he i h paricle, Xi is he posiion vecor of he i h paricle, Ii is he momen of ineria ha equals o 0.4miRi 2 and he roaed angle of paricle i is represened by θi. The symbol 3

4 Fi in Eq. (1) is he resulan conac force generaed by wo collided paricles, i and j. This force can be decomposed furher ino wo componens: one is conac force in normal direcion F ij n and he oher is conac force in angenial direcion F ij, as shown in Eq. (3). The symbol Ti in Eq. (2) is he resulan orque acing on he i h paricle. I can also be decomposed ino wo componens: orques caused by rolling fricion and angenial force, respecively, as given in Eq. (4). The wo conac models adoped in his work are boh hisory deformaion dependen. The difference is from he relaionship beween deformaion and conac force. Gran-Herz-Hisory model describe a nonlinear relaionship beween conac force and overlap disance, while Gran- Hooke-Hisory model gives a linear relaionship. The open-source sofware package LIGGGHTS provides boh of hese models. However he Gran-Herz-Hisory model is modified o include van der Waals force, which can be significan for small paricles, and hereby refer o as Modified Gran-Herz-Hisory model. The normal conac force F ij n can be deermined by [20-22], n F ij K n ξn γ n ij. v nij n ij (5) where in Modified Gran-Herz-Hisory model, parameers are given by 4 5 ln e Kn Yeff Rξn, γn 2 eff Sm n eff, eff, S n 2Yeff Rξn. ln e and in Gran-Hooke-Hisory model, R 2 n eff eff ch K Y m v , γ n 4m K 1 2 ln eff n 2 and v ij represens he velociy of he paricle i relaive o velociy of he paricle j, nij is he uni vecor poin from paricle i o paricle j, e is he coefficien of resiuion of he paricles, R R R / R R is he effecive radius ha represen he geomeric mean diameer of he i and j paricle, i j i j Y eff 1σ 1σ 1/ Y1 Y e. is he effecive Young s modulus ha is calculaed in erms of individual Young s modulus and Poisson raio accordingly, ξn Ri R j Rij is he overlap in normal direcion and m eff mm i j is he effecive masses of he paricles. Characerisic velociy m m i j v ch is aken as uniy in Gran-Hooke-Hisory model. The conac force in angenial direcion is calculaed by [23], n F ij min μ ij, K ij γ ij F ξ v ij (6) where in Modified Gran-Herz-Hisory model, parameers are deermined by, K G eff 5 8Geff Rξ n, γ 2 eff Sm eff, S 8 Rξ 6 G, eff n 22 σ 1 σ 2 2 σ 1 σ 1/ Y1 Y

5 and in Gran-Hooke-Hisory model, K ξ 0 K, γ γ n vd represens he angenial displacemen vecor beween he wo spherical paricles, v [ v v ] ω R ω R is he angenial relaively velociy, ij is he uni vecor i j ij ij i i j j along he angenial direcion, 0 is he ime when he wo paricles jus ouch and have no deformaion, is he ime of collision, ωi or ωj is he angular velociies of paricles i or j and Ri or Rj is he vecor running from he cener of paricle i or j o he conac poin of he wo paricles. The cohesive force is included in boh Modified Gran-Herz-Hisory model and Gran-Hooke- Hisory model. For he cohesive force, Johnson-Kendall-Robers (JKR) model [24] based on Herz elasic heory is used o esimae he cohesive behavior of he paricles. In Herz elasic heory, he normal pushback force beween wo paricles is proporional o he area of overlap beween he paricles. Based on Herz elasic assumpion and meanwhile considering he conac surface as perfecly smooh, he JKR model here is saisfacorily accurae o deermine he cohesive force. In fac, he basic idea is ha if wo paricles are in conac, i adds an addiional normal force ending o mainain he conac, F ka (7) where k is he surface energy densiy and A is he paricle conac area. For sphere-sphere conac [25], conac area A is evaluaed by, ( dis Ri R j )( dis Ri R j )( dis Ri R j )( dis Ri R j ) A (8) 2 4 dis where dis is he cenral disance beween he i and j paricles. Ri and Rj are he radius of he i h and j h paricle, respecively. The van der Waals forces among paricles are included only in he Modified Gran-Herz-Hisory model. The van der Waals force, F v ij beween paricles i and j is given by [26], Ri Rj h Ri R j 2 2 h Rih R jh h Rih R jh Ri R j 5 n v Ha F ij 2 2 (9) where Ha is he Hamaker consan, and h is he separaion of surfaces along he line of he ceners of paricles i and j. A minimum separaion disance h min is considered o preven F v ij becoming infiniy when h goes o zero. The Hamaker consan is relaed o he surface energy densiy by [27]: 2 Ha 24 khmin (10) The orque due o angenial conac force and he orque due o rolling fricion are calculaed in he same way for boh models [28]: T R F (11) T ij i ij ω r ij ij ij μrrk n ξn ij ωij where ω ij ω i ω j is he relaive angular velociy. Table 1 shows he maerial properies and oher physical coefficiens used in hese packing simulaions. The maerial properies of he paricles are same as hose for iron. The surface energy densiy is calculaed from he Hamaker consan. The maerial properies for he conainer is same as ha for he paricles. Table 1 Values of he parameers used in he simulaion process (12)

6 Parameers Values Paricle densiy ρ 7870 kg/m 3 Young s modulus Y N/m 2 Resiuion coefficien e 0.75 Sliding fricion coefficien μs 0.42 Rolling fricion coefficien μr Poison raio σp 0.29 Hamaker consan, Ha J Minimum separaion disance, hmin m Surface energy densiy, k J/m 2 (a) Paricles a = sec Figure 1 (b) Paricles a = 0.2 sec Iniial and final srucure for Gaussian paricles from Modified Gran-Herz- Hisory model wih cohesion. For each simulaion, 4,500 paricles are seled in a simulaion box having lengh and widh equal o 0.006m and he paricles have no iniial physical conac among hem. The iniial porosiy is kep consan a Figure 1 shows he iniial sae of Gaussian paricle packing. As he simulaion ime increases, he paricles begin o fall down due o graviy and hen collide wih oher paricles or wih he boundaries. In his work, all six sides of he simulaion box are considered as physically saionary. Considering he fac ha he conac force is mainly relaed o he paricle deformaion, he ime-sep mus be sufficienly small o preven any unrealisic overlap [29]. In his work, he ime sep is se o be s for all simulaion cases. I should be poined ou ha he velociy of each paricle will hardly reach zero compleely bu he magniude of velociy will approach o an exremely small value. In his work, he paricles are considered o be compleely saionary when heir mean velociies are below m/s. The resuls are presened hrough hree parameers which are widely used o measure he packing srucure: (1) radial 6

7 disribuion funcion (RDF), (2) porosiy ha is he raio of oal volume of void space o he volume aken by all he paricles, and (3) coordinaion number ha is defined as he number of paricles ha are conacing wih he one chosen as reference cener. Resuls and Discussions Sixy scenarios are sudied in his work: five differen mean radius (75µm, 85 µm, 100 µm, 110 µm, and 120 µm) and hree differen size disribuions (mono-sized, uniform and Gaussian) for wo conac models (Modified Gran-Herz-Hisory model and Gran-Hooke-Hisory model) wih and wihou cohesion. The resuls are presened in he form of porosiy, coordinaion numbers, RDF and force disribuion, when he all paricles are compleely packed. Forces considered in he packing process include conac force, which is decomposed ino normal and angenial componens, viscoelasic and fricional forces generaed when collision occurs, graviy which drives he paricles o fall down, cohesive force and van der Waals force which are considered as exernal forces acing on hemselves. Figures 1-4 show he iniial and final packing srucures for hese hree disribuions wih mean diameer of 75 µm. I should be noed ha he deformaion calculaion is very imporan for packing simulaion since he oversimplified model of calculaing overlap disance is always he main reason ha leads o he simulaion crash by inroducing unrealisic energy. Two basic rules are applied o hese packing simulaions: one is ha paricles are always considered as rigid body even hough a deformaion is considered by he chosen model, and he oher is ha he criical cenral disance is se for paricle deformaion. The criical disance is 1.01(d1+d2)/2 where d1 and d2 are he diameers of he wo paricles. I means when he cenral disance of wo paricles is less han he criical disance he wo paricles are considered o be in direc conac [6]. (a) Paricles a = sec (b) Paricles a = 0.2 sec Figure 2 Iniial and final packing srucure for mono-sized paricles from Modified Gran- Herz-Hisory model wih cohesion. 7

8 (a) Paricles a = sec Figure 3 (b) Paricles a = 0.2 sec Iniial and final packing srucure for uniform size paricles from Modified Gran- Herz-Hisory wih cohesion. (a) Paricles a = sec Figure 4 (b) Paricles a = 0.2 sec Iniial and final srucure for Gaussian paricles from Gran-Hooke-Hisory model wih cohesion. 8

9 (a) Modified Gran-Herz-Hisory Model Figure 5 (b) Gran-Hooke-Hisory Model Effec of porosiy wih paricle size and disribuion Figures 5 and 6 presen he porosiies and coordinaion numbers for differen cases. I can be seen ha he porosiy decreases along wih he increasing paricle radius for all disribuions when cohesive forces are considered. Similar rend was observed in he work of previous researchers [30]. This decrease in porosiy wih increase in radius is expeced since wih increase of radii or masses of he paricles he iniial supplied energy (graviaional poenial) also increases. So he effec of cohesion in he packing of paricles decreases and he porosiy values become closer o ha for Random Loose Packing [2, 31]. This also explains he decrease in differences beween differen size disribuions in erms of porosiy when he radius increases. Among he hree disribuions considered, Gaussian disribuion has he highes porosiy and mono-size has he lowes. The porosiy values for he wo models, Modified Gran-Herz and Gran-Hooke are slighly differen bu boh show he same rend. As for he non-cohesion case porosiy also decreases wih increase in paricle radius, bu he porosiy values are much smaller. Figure 5 also shows ha he rae of decrease of porosiy wih radius for noncohesion case is much smaller. For mono-sized disribuion wihou cohesion, porosiy remains almos consan for boh Modified Gran-Herz-Hisory model and Gran-Hooke-Hisory model. Since here is no cohesion he dissipaive forces are smaller and paricles can pack more closely. Again he difference beween he wo models in non-cohesion cases is very small. For he coordinaion number, he rends 9

10 for hree disribuions wih cohesion are similar. I can be observed ha he coordinaion number increases as paricle radius increases which is exacly he opposie of he rend of porosiy. Unlike porosiy, Gaussian disribuion now has he lowes coordinaion number and mono-size disribuion has he highes. Ineresingly, i is found ha here is no significan change in coordinaion number wheher or no cohesion is included. However one can expec ha coordinaion number should be smaller when here is no cohesion (porosiy is larger). This can be explained as follows. When here is cohesion, paricles end o clump ogeher and form clusers. These clusers have void spaces in hem. Due o his formaion of clusers in some region paricles have high coordinaion number and in some region he coordinaion number is small. The coordinaion numbers given in Tables 2-5 and Figure 6 are average of coordinaion numbers for all paricles. I can be seen ha he coordinaion numbers for cohesion and non-cohesion cases are similar. (a) Modified Gran-Herz-Hisory Model Figure 6 (b) Gran-Hooke-Hisory Model Effec of coordinaion number wih paricle size and disribuion 10

11 Table 2 Porosiy and coordinaion number for Modified Gran-Herz-Hisory model Table 3 Porosiy and coordinaion number for Gran-Hooke-Hisory model Table 4 Porosiy and coordinaion number for Modified Gran-Herz-Hisory model wihou cohesion Table 5 Porosiy and coordinaion number for Gran-Hooke-Hisory model wihou cohesion Porosiy Coordinaion number Radius Monosizesized Mono- 75μm μm μm μm μm Porosiy Coordinaion number Radius Monosizesized Mono- 75μm μm μm μm μm Porosiy Coordinaion number Radius Monosizesized Mono- 75μm μm μm μm μm Porosiy Coordinaion number Radius Monosizesized Mono- 75μm μm μm μm μm

12 (a) Modified Gran-Herz-Hisory wih cohesion (b) Modified Gran-Herz-Hisory wihou cohesion (c) Gran-Hooke-Hisory wih cohesion (d) Gran-Hooke-Hisory wihou cohesion Figure 7 RDF for paricles wih 75 μm radius. (a) Modified Gran-Herz-Hisory wih cohesion (b) Modified Gran-Herz-Hisory wihou cohesion 12

13 (c) Gran-Hooke-Hisory wih cohesion (d) Gran-Hooke-Hisory wihou cohesion Figure 8 RDF for paricles wih 85μm radius. (a) Modified Gran-Herz-Hisory wih cohesion (b) Modified Gran-Herz-Hisory wihou cohesion (c) Gran-Hooke-Hisory wih cohesion (d) Gran-Hooke-Hisory wihou cohesion Figure 9 RDF for paricles wih 100 μm radius. 13

14 (a) Modified Gran-Herz-Hisory wih cohesion (b) Modified Gran-Herz-Hisory wihou cohesion (c) Gran-Hooke-Hisory wih cohesion (d) Gran-Hooke-Hisory wihou cohesion Figure 10 RDF for paricles wih 110μm radius. (a) Modified Gran-Herz-Hisory wih cohesion (b) Modified Gran-Herz-Hisory wihou cohesion 14

15 Figure 11 (c) Gran-Hooke-Hisory wih cohesion RDF for paricles wih 120μm radius. (d) Gran-Hooke-Hisory wihou cohesion Figures 7-11 show he RDF for paricle sysems wih mean radius of 75 µm, 85µm, 100 µm, 110 µm and 120 µm and associaed wih hree differen size disribuions (mono-sized, uniform and Gaussian). For he cases where he paricles have he same radius, hree main apparen peaks appear. The firs peak is sharply a 2r which is for he iniial one o one conac, he second and he hird are a around 2 2r and 4r, respecively which corresponds o he wo characerisic paricle conac ypes, namely edge-sharing-in-plane equilaeral riangle and hree paricles ceners in a line (he hree conac ypes are illusraed in Figure 7 (a)). The second and hird peaks merge ino a single second peak for oher disribuions. The paricle sysems wih mono-size disribuion usually have he highes peak values among all hree cases. The peak values for Herz model are close o ha for Hooke model. Also he peak values of RDF are almos same for cohesion and noncohesion cases. (a) Modified Gran-Herz-Hisory wih cohesion (b) Modified Gran-Herz-Hisory wihou cohesion 15

16 (c) Gran-Hooke-Hisory wih cohesion (d) Gran-Hooke-Hisory wihou cohesion Figure 12 Force disribuion for paricles wih 75 μm radius and Gaussian disribuion. (a) Modified Gran-Herz -Hisory wih cohesion (b) Modified Gran-Herz-Hisory wihou cohesion (c) Gran-Hooke -Hisory wih cohesion (d) Gran-Hooke -Hisory wihou cohesion Figure 13 Force disribuion for paricles wih 75 μm radius and mono-size disribuion. 16

17 (a) Modified Gran-Herz -Hisory wih cohesion (b) Modified Gran-Herz -Hisory wihou cohesion (c) Gran-Hooke-Hisory wih cohesion (d) Gran-Hooke-Hisory wihou cohesion Figure 14 Force disribuion for paricles wih 75 μm radius and uniform disribuion. Figure show he force disribuion resuls afer he paricles are compleely packed for paricle sysems wih mean radius of 75µm. The force disribuion graphs for oher paricle radii look similar. For same disribuion he couns for each force magniude are no exacly he same bu close. However as he paricle radius increases, he force magniudes increase as a response. Tables 6 and 7 give he mean ne force for all he cases when he paricles are finally packed. I has o be poined ou ha he resulan force here does no represen graviy, since effec of graviy is small (of he order of N). I can also be observed ha he ne force does no vary much even when he cohesion is included. The mean ne force increases wih he size of he paricles, and i can also be seen ha his force has he larges value if he paricle size follow uniform disribuion. Paricles wih Gaussian disribuion have he secondary magniude of force, while he mono sized paricles have he smalles ne force. The difference in magniude of mean ne force beween he wo differen conac models is negligible. By comparing he wo models, Modified Gran-Herz-Hisory and Gran-Hooke-Hisory, i can be seen ha he difference beween hem is no significan in erms of porosiy, coordinaion number and mean ne force. Boh of hese models assume ha he paricles are viscoelasic and have a siffness erm and dissipaion erm. As poined ou by [32] he linear Gran-Hooke model can be as accurae as he non-linear Modified Gran-Herz model if he siffness consans, Kn and K, and damping coefficiens, γn and γ, are evaluaed carefully. In his sudy, even hough cohesion is included, he resuls obained from he wo models are sill close. Van der Waals force included in he Modified Gran Herz model did no seem o play a grea role in he packing process. This 17

18 migh be because paricle sizes are oo large for van der Waals force o ake effec. When he efficiency of he wo models are considered, he simulaions wih he Gran-Hooke-Hisory model ran faser han he simulaion wih he Modified Gran-Herz-Hisory model. So he linear Gran- Hooke-Hisory model is more efficien han he Modified Gran-Herz-Hisory model. Table 6 Magniude of mean ne conac force (N) for Modified Gran-Herz-Hisory model Cohesion No Cohesion Radius Monosizesized Mono- 75μm μm μm μm μm Table 7 Magniude of mean ne conac force (N) for Gran-Hooke-Hisory model Cohesion No Cohesion Radius Monosizesized Mono- 75μm μm μm μm μm Conclusions A sudy on packing srucures of paricle sysem wih differen radii and size disribuions using wo differen models are carried ou by he Discree Elemen Mehod. The simulaion resuls including RDF and force disribuion, porosiy and coordinaion number are presened. I was observed ha he paricles wih Gaussian disribuion always have he lowes packing densiy while he paricles wih uniform size disribuion have he medium packing densiy and mono-sized paricles normally have he highes packing densiy. For he paricles packing under cohesive effec, size disribuions resul in he same endency of packing densiy bu has much less variaion wih paricle size. Coordinaion number is no affeced by cohesion significanly bu paricle size and size disribuion do influence he resul. The differences in porosiy, coordinaion number, RDF and magniude of mean ne force beween he wo models used are no subsanial which show ha any of he models can be used for simulaion of paricle packing. However when efficiency is considered he Gran-Hooke-Hisory model is found o be more efficien han he Modified Gran- Herz-Hisory model. So Gran-Hooke-Hisory model can be he model of choice for simulaing micro-sized paricles. Acknowledgemen Suppor for his work by he U.S. Naional Science Foundaion under gran number CBET is graefully acknowledged. 18

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