Physics of Compaction of Fine Cohesive Particles

Physics o Compac ion o Fine Cohesi e Pa icles A. Cas ellanos, J. M. Val e de, and M. A. S. Quin anilla Depa amen o de Elec onica y Elec omagne ismo, Uni e sidad de Se illa, A enida Reina Me cedes s/n, 41012 Se illa, Spain (Recei ed 9 No embe 2004; published 22 Feb ua y 2005) Fluidized ac al clus e s o ine pa icles display c i ical-like dynamics a he jamming ansi ion, cha ac e ized by a powe law ela ing consolida ion s ess wi h olume ac ion inc emen [ ^c/]. A a c i ical s ess clus e s a e dis up ed and he e is a c osso e o a loga i hmic law (log ^c) esembling he phenomenology o soils. We measu e @1==@ log ^c/Bo0:2 g, whe e Bogis he a io o in e pa icle a ac i e o ce (in he luidlike egime) o pa icle weigh . This law sugges s ha compac ion is uled by he in e nal packing s uc u e o he jammed clus e s a nea ly ze o consolida ion. DOI: 10.1103/PhysRe Le .94.075501 PACS numbe s: 61.43.G , 45.70.Cc, 61.43.H , 81.20.E Empi ical s udies on he compac ion o soils da e back o he beginning o he las cen u y. Walke [1] i ed his da a by he loga i hmic law 1=logc=c0, whe e is he pa icle olume ac ion, c he applied consoli- da ion s ess, and (comp ession index) and c0a e em- pi ical pa ame e s. This equa ion applies well in loose samples, whe e compac ion is d i en by ea angemen o pa icles, and has been adi ionally used in ci il engi- nee ing [2,3]. An essen ial ing edien in mos g anula sys- ems is cohesion. Tes s on cohesi e powde s show ha  dec eases wi h he pa icle olume ac ion o he ini ial s a e [4,5], indica ing ha in e pa icle a ac i e o ces, which a o he o ma ion o po ous s uc u es, play a ele- an ole in he compac ion p ocess. Ye he ini ial s a e in ypical enginee ing expe imen s in ol es consolida ion s esses c0>10 kPa [5]. Many indus y applica ions de- mand esea ch on smalle consolida ions as hese co e- spond o condi ions o powde low. Fo example, in he handling o xe og aphic one s, ypical consolida ions ange om a ew pascals o a ew hund ed pascals. Mo eo e , expe imen s a low consolida ions ha e a undamen al in- e es in o de o cha ac e ize he ansi ion om he luid- like o he solidlike s a e (jamming) [6] since he s uc u al p ope ies o he unconsolida ed jammed s a e (c’0), which is he uly ini ial s a e in any compac ion p ocess, a e de e minan on he ea angemen o he u he loaded pa icles. We s udy he compac ion o ine pa icles wi h con olled a ac i e o ce, ini ially luidized and la e sub- jec ed o loads om jus a ew pascals up o 10 kPa. Ou no el expe imen al s udy is aimed o shed ligh on he ole o he ini ial s a e, i.e., he unconsolida ed jammed s a e, on compac ion. The powde s es ed a e xe og aphic one s based on polyme (pa icle densi y p’1g=cm3). They a e p oduced by an a i ion p ocess, hus ha ing an i egu- la shape, and size classi ied in a ange o pa icle sizes (dp) om 19.1 o 7mby ae odynamic classi ica ion, showing a na ow pa icle size dis ibu ion (see Fig. 1). Addi ionally, he powde s a e blended wi h umed silica nanopa icles (ei he 8 o 40 nm nominal diame e s) o coa uni o mly he polyme pa icle su ace in concen a ions om 10% o 100% o su ace a ea co e age (SAC). In he luidized egime he e is an a ac i e o ce F0 be ween he d y and uncha ged pa icles mainly a ising om he an de Waals in e ac ion F0F dW ’Ada= 24z2 0, whe e z0’4 Ais he dis ance o closes app oach be ween wo molecules, Ais he Hamake cons an , and da is he ypical size o he su ace aspe i ies ( ypically A 1019 Jand da0:2m) [7]. An es ima ion o elec o- s a ic o ces om cha ge spec og aph measu emen s shows ha hey a e much smalle han an de Waals o ces as commonly accep ed in he li e a u e o elec oneu al ine powde s [7,8]. A d y ni ogen a mosphe e minimizes also capilla y o ces. Because o he s ong in e pa icle a ac i e o ce as compa ed o pa icle weigh , one pa - icles a e clus e ed in he luidlike egime [9]. Acco ding o ou p e ious expe imen al esul s he ypical numbe o pa icles pe clus e Nand ypical a io o clus e size o pa icle size depend on he a io o a ac i e o ce o pa icle weigh F0=mpgBog(g anula Bond num- be ). In pa icula we ound NBo g(’0:7) and D lnN=ln’2:5 o he ac al dimension, in ag eemen FIG. 1. Typical pic u e om he op ical mic oscope o one pa icles clus e ed in a nonaqueous liquid suspension ha eminds one o a di usion-limi ed agg ega e. Fo his one (12:7mpa icle size and 10% SAC) he ac al dimension ob ained om se ling expe imen s in gas luidiza ion [9] is D’2:53. PRL 94, 075501 (2005) PHYSICAL REVIEW LETTERS week ending 25 FEBRUARY 2005 0031-9007=05=94(7)=075501(4)$23.00 075501-1 2005 The Ame ican Physical Socie y wi h he di usion-limi ed-agg ega ion (DLA) model p e- dic ion. The ypical size o su ace aspe i ies a con ac can be dec eased down o he size o silica agglome a es co e - ing he pa icle su ace o la ge enough SAC, hus educ- ing F dW. (F om scanning elec on misc osopy mic o- g aphs we obse e silica agglome a es o ypical size da’ 50 nm o 8 nm silica nanopa icles and da’200 nm o 50 nm silica nanopa icles.) Since ou ine pa icles a e a he mal, he limi s o clus e g ow h in he ini ial luidized s a e esul om he in e play o g a i a ional and low shea e ec s [10]. In o de o es ima e he limi size o ou DLA clus e s we will adop a simila app oach o ha used in Re . [11], whe e he limi s o gela ion in colloidal agg ega ion we e s udied. In he luidized bed he weigh o he clus e is balanced by he hyd odynamic d ag om he su ounding gas. F ac al clus e s sc een ex e nal ields e y e ec i ely and he luid low inside he clus e is negligible compa ed o he low ou side, hus he d ag ac s mainly a he su ace o he clus e whe eas g a i y is a body o ce ac ing uni o mly h ough he clus e . This esul s in shea o ces dis ibu ed ac oss he clus e limi ing i s size. Using a simple sp ing model o he clus e , i has been shown [10,11] ha he ypical s ain on he clus e is Nmpg=KcRc, whe e Kcis he clus e sp ing cons an and Rcdp=2is he clus e adius. Kcis gi en by k0=, whe e k0is he in e pa icle o ce cons an , and he elas ici y exponen is 3in he 3D case [12]. Thus he local shea o ce inside he clus e is Fsk0dp=2 mpgD2. Manley e al. [11] use a c i ical alue, mea- su ed independen ly, o he maximum s ain sus ainable o calcula e he maximum size o hei agg ega es. Mo e gene ally, we may es ima e ha he c i ical shea o ce mus be o o de o he in e pa icle a ac i e o ce Fmax s F0, which leads o BogD2, hus he maximum numbe o pa icles pe clus e should be NDBoD=D2 g. Fo DLA clus e s (D2:5) we ob ain NBo0:6 g, in close ag eemen wi h ou p e ious expe imen al esul s [9]. Now we can explain why, o a cons an F0(cons an SAC), he size o ou clus e s measu ed in Re . [9] was weakly dependen on pa icle size since he c i e ion p edic s kdp/d0:3 p. Mo eo e , o ou ypical clus e s (<10) he in e clus e Bond numbe is Bo gBog=N 2< 100, i.e., in e clus e cohesi eness is small. In he expe imen al pa o ou wo k we use he luidized bed es e o measu e as a unc ion o c. A de ailed epo abou he unc ioning o his appa a us can be ound in Re . [13]. A d y ni ogen se es o con ol c, being pumped upwa d o downwa d h ough he powde bed while he gas p essu e d op pac oss he bed is ead om a di e en ial p essu e ansduce . is de i ed om he heigh o he bed, which is measu ed by means o an ul asonic senso . In o de o subjec he powde o e y low s esses, like in mic og a i y, he bed is allowed o se le unde a small upwa ds di ec ed gas low. In his way cis lowe ed down o cWp, whe e Wis he powde weigh pe uni a ea and pinc eases as he alue o he decomp essing gas low used is la ge . In Fig. 2 we ha e plo ed nea he jamming ansi ion as a unc ion o c o one Canon CLC700 (100% SAC, in his comme - cial one he addi i e is TiO2) and o an expe imen al one wi h simila pa icle size (7:8m) bu only 32% SAC. The jamming ansi ion is discussed in de ail in a p e ious wo k [14]. As seen in he da a depic ed in Fig. 2, i was gene ally obse ed ha in a ange o e y small s esses he  s c ela ionship ollows a c i ical-like unc ional o m c/J, p edic ed by simula ions [15] and eminiscen o an equilib ium c i ical phenomena. In ou new expe imen al s udy consolida ion s esses la ge han Wa e applied by a downwa ds di ec ed gas low. cis hus inc eased up o cWp. We see in Fig. 2 ha he inc emen o as cis inc eased de ia es om he c i ical-like powe law, and a a c i ical s ess c010 Pa c osses o e o a loga i hmic law ’J log ^c(^cc=c0) ema kably simila o he empi i- cal equa ion usually employed o desc ibe he compac ion o g anula ma e ials such as soils [2] in he ea angemen egime. Likely a ^c’1clus e s ha e eached hei closes andom packing ( RCP). The c osso e o he loga i hmic law occu s o olume ac ions o clus e s smalle han 0.64 ( andom close packing o noncohesi e ha d sphe es) as i migh be expec ed om he exis ence, al hough small, o in e clus e cohesi eness. Fo example,  RCP ’0:53  FIG. 2. Pa icle olume ac ion as a unc ion o he consoli- da ion s ess. Da a ob ained by consolida ing he powde allow- ing i o se le unde upwa d di ec ed gas lows (solid sym- bols) a e join ly plo ed wi h da a ob ained by consolida ing he powde by means o a downwa d di ec ed gas low (open symbols). Da a co espond o comme cial one Canon CLC700 (100% SAC) and o an expe imen al one wi h educed su ace addi i e co e age (32% SAC) and simila pa icle size. The lines co espond o he powe law c/J ha i s o he da a in a ange c&c010 Pa and o a loga i hmic i J’ logc=c0 ha i s o he da a in he ange c*c0. Inse : da a in he loga i hmic egion o one s wi h he same SAC (32%) bu di e en pa icle size (indica ed). PRL 94, 075501 (2005) PHYSICAL REVIEW LETTERS week ending 25 FEBRUARY 2005 075501-2 0:02 o clus e s o p ima y pa icle size om 7.8 o 19:1mand 32% SAC (see Fig. 4 in Re . [14]), ma ching he epo ed alue [16] o sphe es o equi alen size o hese clus e s (kdp’50 m). The inse o Fig. 2 shows da a o hese one s wi h he same SAC (32%) bu di e - en pa icle size. I is obse ed ha ’0:04 is almos independen o pa icle size, sligh ly dec easing when he SAC is inc eased (see main g aph). Thus he co ela ion be ween and Bogseems o be a leas a second o de e ec (no e ha a dec ease o dp om 19.1 o 7:8m co e s a wide ange o Bog). Howe e , he da a epo ed in he enginee ing li e a u e e eal clea co ela ions be ween @1==@ log ^cand he pa icle olume ac ion o he lowes consolida ion s a e. We plo in he inse o Fig. 3 he da a o he same one s (32% SAC) o 1= s ^c, which a e also well i ed o a loga i hmic law 1= ’ 1=Jlog ^c. The main g aph shows as a unc ion o Bog(calcula ed assuming F0F dW). A powe law ’0:1Bo0:21 gis clea ly seen. The ex apola ed alue o Bog1(’0:1) ma ches he ypical alue epo ed o noncohesi e g anula ma e ials (Bog&1) such as sand [3]. Wha is he physical o igin o his law? In a i s o de app oach we may app oxima e 1=’ @1==@ log ^cJlog ^c’1=J2log ^c, whe e ’@=@ log ^cJ. Thus ’1=J2. Le us w i e J’ RCPc, whe e cN=3D3is he pa - icle olume ac ion wi hin each clus e . Then ’ 1= RCP2Bo62D=D2 g1= RCP2Bo0:22 g, whe e we ha e used he clus e size limi c i e ion (Bog D2), and D2:5. Clus e s beha e as low cohesi e e ec i e sphe es and hus  RCP will be almos independen on Bog. Using ’0:04 and  RCP ’0:53 o he one s wi h 32% SAC, we would p edic ’0:14Bo0:22 g, in good ag eemen wi h he expe imen al esul . We plo in Fig. 4 he da a o  s Bog o o he one s wi h SAC >30% o which we admi F0F dW. The new da a also scale wi h Bogin acco dance wi h he p edic ed law. In summa y, his law emphasizes he undamen al ole o size and ac al s uc u e o he jammed clus e s on he dis ibu ion o oids o be illed in he compac ion p ocess. The case o highly cohesi e powde s (SAC <30%) needs, howe e , addi ional discussion. In he inse o Fig. 4 we include da a o highly cohesi e one s wi h only 20% SAC using F0 F dW o calcula e Bog. The pa icle size o hese one s is ’7mand he base polyme is polyes e ea ed wi h di e en amoun s o a c oss-linking agen (gel) ha p o- duces a sligh inc ease o he polyme ha dness (in any case small compa ed wi h he e ec o silica). The da a de ia e clea ly om he scaling law, showing unexpec edly la ge alues o  ha mus indica e he exis ence o la ge clus- e s. Mo eo e , in spi e o he simila alues o Bog(same pa icle weigh and same an de Waals o ce: A’1019 J, FIG. 3. Comp ession index e sus he g anula Bond numbe o one s wi h he same su ace addi i e co e age (32% SAC) and a ying pa icle size (indica ed). The con inuous line is a powe law i o he da a (’0:1Bo0:21 g). Inse : in e se o pa icle olume ac ion e sus he consolida ion s ess, whe e lines a e loga i hmic i s o he da a (c010 Pa). FIG. 4. Comp ession index e sus g anula Bond numbe o one s wi h di e en pa icle size and % SAC (indica ed). In he main g aph Bogis calcula ed assuming ha he in e pa icle a ac i e o ce in luidiza ion is he an de Waals o ce o SAC >30%, while con ac memo y (see ex ) is conside ed o one s wi h SAC <30%. In he inse he an de Waals o ce is used o all he one s. Fo one s wi h 20% and 100% SAC, he e ec o inc easing he size o silica nanopa icle addi i es om 8 o 50 nm has been es ed. Addi ionally, ou di e en amoun s o c oss-linking agen in he pa en polyme ( om 0% o 45%) ha e been used in hese one s. The con inuous lines a e powe law i s o he da a, only o one s wi h SAC >30% in he inse (/Bo0:24 g) and o he whole se o one s in he main g aph (/Bo0:25 g). PRL 94, 075501 (2005) PHYSICAL REVIEW LETTERS week ending 25 FEBRUARY 2005 075501-3 da’0:2m), he e is a clea di e ence be ween he comp ession indexes o one s wi h di e en gel con en . Silica addi i es imp o e powde lowabili y mainly by inc easing he local ha dness Hon he con ac as i is p o en by he be e lowabili y o one s blended wi h 200 nm silica agglome a es, o simila size o he ypical polyme aspe i y size. The s ong a ac i e o ces be ween loaded ine pa icles cause plas ic de o ma ion o con ac s, leading o a ele an inc ease o he adhesi e o ce wi h he applied load [17]. Tone s ei he wi hou o wi h low pe - cen age o he ha d silica addi i e ha e a e y poo low- abili y because i is di icul o b eak in e pa icle con ac s be ween p e iously loaded pa icles, i.e., he in e pa icle con ac s in highly cohesi e powde will p ese e he mem- o y o he ini ial loaded s a e. I is likely ha agmen s ha ea lie exis ed as agg ega es in he loaded powde pe sis in luidiza ion gi ing ise o la ge clus e s o s ongly adhe ed pa icles (a simila phenomenon has been ecognized in agmen a ion o colloidal suspensions o s ongly cohe- si e pa icles [18]). Thus he a ac i e o ce be ween clus e ed pa icles in luidiza ion o hese powde s mus be much la ge han he an de Waals o ce. The inc eased con ac ha dness by silica addi i e educes he a e o inc ease o he adhesi e o ce wi h load [17], hus allowing o an easy b eaking o in e pa icle con ac s by an ex e nal ene gy sou ce (such as gas luidiza ion), and he e o e imp o ing lowabili y. Fo hese low cohesi e one s he an de Waals o ce was indeed a good app oxima ion o he in e pa icle a ac i e o ce in luidiza ion. Fo one s wi h only 20% SAC many con ac s a e be ween polyme su aces. The a e o inc ease o he adhesi e o ce wi h load inc eases as His inc eased [17], which means ha con ac s wi h smalle ha dness will gi e ise o la ge clus e s and hus o la ge alues o . (This explains he e ec o gel seen on .) The p oblem is, How can we es ima e Bog o hese highly cohesi e one s wi h ha d- ness dependen memo ies? One possibili y is o use he c i e ion o clus e limi size in luidiza ion, BogD2, whe e D’2:5acco ding o sedimen a ion es s [9], and  may be ob ained om he in o ma ion on he ini ial jammed s a e: D=3J= J. In his equa ion he mos impo an and a iable pa ame e is J, which is accu a ely ob ained om he loga i hmic i equa ion J log ^c, while  Jcan be expec ed o show a small a ia ion be ween 0:5(10% SAC) and 0:55 (abo e 60% SAC). Fo low cohesi e one s (SAC *30%), he es ima ed Bogin his way is simila o he p e iously calcula ed one assuming F0F dW, bu , o he highly cohesi e ones, Bogis signi ican ly la ge as we an icipa ed. The main g aph o Fig. 4 includes he da a o 20% and 10% SAC one s wi h he new es ima ion o Bog, showing a good i o he p edic ed powe law. We can also dis- c imina e now be ween he comp ession indices o 20% SAC one s in he base o he memo y e asing e ec o he c oss-linking agen added o he pa en polyme . To conclude, we ha e in es iga ed he compac ion be- ha io o cohesi e pa icles which a e clus e ed in he luidlike egime. The numbe o pa icles in ou DLA clus e s, and hus he clus e packing ac ion (c), is con olled by he a io o he in e pa icle a ac i e o ce o pa icle weigh Bog(c’Bo0:1 g). In he close icini y o jamming and abo e a c i ical s ess c010 Pa, he e is a c osso e o he loga i hmic law 1= ’1=J logc=c0. Expe imen al s udies in he enginee ing li e a u e ha e usually shown his beha io , and he com- p ession index has been co ela ed o he minimum pa icle olume ac ion (co esponding o he smalle s ess applicable, ypically 10 kPa); he la ge i is, he la ge he . Ou luidiza ion echnique allows o a s udy o compac ion beha io om jus a ew pascals, hus we a e able o ela e  o he uly ini ial s a e o ea ange- men . The ini ial dis ibu ion o oids o be illed, which ules he compac ion p ocess, is mainly de e mined by he in e nal packing s uc u e o clus e s ha a e jammed a he ini ial unconsolida ed s a e. In a i s o de analysis we es ima e /1=c J2’Bo0:2 g, in ag eemen wi h ou mea- su emen s. Ou analysis implies also ha , due o he high plas ici y o in e pa icle con ac s in highly cohesi e pow- de s, hese powde s mus e ain memo y in luidiza ion o p e ious loaded s a es; when jammed, e y la ge clus e s p oduce e y po ous ini ial s a es and as a consequence la ge alues o he comp ession index. We acknowledge he Xe ox Founda ion and he Spanish Minis e io de Ciencia y Tecnologia (Con ac No. BMF2003-01739). [1] E. E. Walke , T ans. Fa aday Soc. 19, 73 (1923). [2] J. A kinson, The Mechanics o Soils and Founda ions (McG aw-Hill, London, 1993). [3] P. E esque, Poud es and G ains 10, 6 (1999). [4] D. Poquillon e al., Powde Technol. 126, 65 (2002). [5] J. H. Pa k and T. Koumo o, J. Geo ech. Geoen i on. Eng. 130, 223 (2004). [6] J. M. Val e de e al., Phys. Re . Le . 86, 3020 (2001). [7] K. Rie ema, The Dynamics o Fine Powde s (Else ie , London, 1991). [8] H. K upp, Ad . Colloid In e ace Sci. 1, 111 (1967). [9] A. Cas ellanos e al., Phys. Re . E 64, 041304 (2001). [10] Y. Kan o and T. A. Wi en, J. Phys. Le . 45, L675 (1984). [11] S. Manley e al., Phys. Re . Le . 93, 108302 (2004). [12] Y. Kan o and I. Webman, Phys. Re . Le . 52, 1891 (1984). [13] J. M. Val e de e al., Re . Sci. Ins um. 71, 2791 (2000). [14] J. M. Val e de e al., Phys. Re . Le . 92, 258303 (2004). [15] C. S. O’He n e al., Phys. Re . E 68, 011306 (2003). [16] R. Y. Yang e al., Phys. Re . E 62, 3900 (2000). [17] M. A. S. Quin anilla e al., Phys. Re . E 64, 031301 (2001). [18] Y. Ta ek e al., Powde Technol. 143–144, 117 (2004). PRL 94, 075501 (2005) PHYSICAL REVIEW LETTERS week ending 25 FEBRUARY 2005 075501-4