Influence of Morphology and Packing Properties on the Permeability of Fine Particle Agglomerates

Transcription

1 Influence of Morhology nd Pcking Proerties on the Permebility of Fine Prticle Agglomertes Alberto Scurti Ic Mns-Zloczower * Det. Mcromoleculr Science nd Engineering Donld L. Feke Det. Mcromoleculr Science nd Engineering Cse Western Reserve University Clevelnd, OH Abstrct When disersing owder gglomertes in olymeric medi, liquid infiltrtion driven by cillry ressure occurs ffecting both gglomerte strength nd hydrodynmic forces trnsmitted to the solid. It is of rmount imortnce to be ble to redict the extent of olymer infiltrtion in owder gglomertes. In this chter we resent exerimentl dt nd model describing the influence of owder morhology nd cking roerties on the ermebility of fine rticle gglomertes. Cillry rise exeriments were erformed to study infiltrtion behvior of olymers into owder comcts with different densities. The infiltrtion curves showed two different regimes which cn be ttributed to chnges in the gglomerte ermebility uon comction. We exlin these regimes in terms of hierrchicl structure for the gglomertes nd different cking behvior t the level of ggregtes nd rimry rticles. Inter-ggregte ores hve lrger size nd llow fster infiltrtion wheres ores within the ggregtes re smller nd show retrded infiltrtion. Using s the void frction between solid (full) ggregtes nd s the void frction inside the ggregtes, the overll orosity,, is clculted from + (1- ) Reworking the Crmn-Kozeny eqution, we identify two min contributions to the overll ermebility, nmely one due to inter-ggregte orosity nd one due to intr-ggregte orosity. This lst term in turn, cn be identified s contrived of two terms: one referring to the ores inside * Corresonding Author e-mil: ixm@o.cwru.edu

2 the ggregtes, the second reflecting the collse of lrge ores uon comction. At low density the first term is redominnt nd leds to fst infiltrtion. Uon comction beyond criticl density, the intr-ggregte orosity tkes the leding role chnging drsticlly the infiltrtion rte. Introduction The utke of liquids by rticle gglomertes due to cillry ction hs vriety of rcticl imlictions. For exmle the disersion of fine rticle gglomertes into liquids is influenced by fluid infiltrtion into the clusters. It is therefore imortnt to guge nd nlyze the extent of fluid infiltrtion during rocessing oertion. Infiltrtion exeriments my be used to robe owder morhology nd mcroscoic roerties relted to their structure. Therefore understnding the reltionshi between cluster ermebility nd the owder roerties is imortnt. Infiltrtion of liquids into orous medi hs been studied by severl reserchers who tried to formulte mcroscoic lws relting the flow field with imortnt owder chrcteristics such s orosity nd rticle size. These studies minly rely on Drcy's lw: k q P (1.1) η correlting the liner flow rte q, with fluid ressure grdient P (with P the cillry ressure due to fluid surfce tension), fluid viscosity η, nd the orous medi ermebility, k. One very well known eqution which reltes the ermebility k to orous medi roerties, ws derived by Kozeny [Scheidegger, 1957]. Kozeny viewed the orous bed s n ssemblge of chnnels of vrious cross sections nd exressed the ermebility s: 3 c (1.) S T k where is the orosity of the orous medi, S the secific surfce of the chnnel, T the tortuosity fctor nd c roortionlity rmeter which deends on the she of the chnnels. The tortuosity fctor tkes into ccount the comlexity of the chnnels in the orous medi. The Kozeny eqution hs been lrgely lied nd lso modified by other reserchers. Crmn introduced the secific surfce exosed to the fluid S 0 (S 0 S(1-)) nd set the constnt c to 1/5

3 which gve the best fit to his exerimentl results [Scheidegger, 1957]. The result is known s the Kozeny-Crmn eqution: 3 k c (1.3) 5S (1 ) 0 A more recent modifiction of the Kozeny-Crmn eqution is due to Blke [Bird, 1960], who relted ermebility to the void frction nd rimry rticle size D nd introduced correction fctor derived from exerimentl results. In this cse the ermebility k is written s: D 3 k c (1.4) 150(1 ) This eqution is considered to be vlid for medi consisting of individul rticles. Dullien nd McDonld ddressed the roblem of multisized rticles resent in orous medi. Dullien [Dullien, 1976] modified the Kozeny-Crmn eqution ssuming ores with eriodic ste chnges in their dimeter. The result is n rent cillry dimeter which is function of ll cillry sizes resent in the orous medi. McDonld [McDonld, 1991] generlized the Blke-Kozeny eqution for multisized shericl rticles showing tht D, in the Blke-Kozeny eqution, hs to be relced by M /M 1, where M nd M 1 re the second nd first moment of the size distribution function for the shericl rticles. These models find good greement with exerimentl dt. Bohin [Bohin, 1994] studied infiltrtion into shericl gglomertes driven by cillry ressure. Strting from the Blke-Kozeny eqution, he develoed quntittive model ble to ccurtely describe the rocess. We try to exlin the different regimes in n infiltrtion rocess brought bout by the comction of owder bed which is used to reresent owder gglomertes. Exerimentl results obtined t high density re well described by the Blke-Kozeny eqution. However, the owder bed ermebility t t density ers to be well bove tht redicted by the theory. In this er we ttemt to exlin these results in terms of hierrchicl structure of the owder bed, resenting ermebility model which mkes distinction between the orosity t the level of rimry rticles nd ggregtes.

4 Exerimentl Mterils Severl owders were tested, nmely reciitted silic, titnium dioxide, clcium crbonte nd crbon blck. These owders rovide vriety of morhologies nd re fillers commonly used in olymer rocessing [Ottino, 000]. The fluids used to study infiltrtion were olydimethylsiloxne (PDMS) liquid olymers. These re very imortnt silicone oils nd smles, snning in rnge of viscosity, re vilble. Selected roerties of the mterils re summrized in Tble 1. Procedures Infiltrtion of liquids into the owder t t density ws erformed in cillry rise exeriments (Wshburn exeriment). The rtus consists of beker contining reservoir of liquid (llowing the ssumtion of n infinite reservoir [Wshburn, 191] nd glss tube suorted by clm. The schemtic of the rtus is resented in Fig. 1. Glss wool lced t the bottom of the tube, llows fluid to ss while suorting the owder. The tube, tht hs volume mrkers (e.g. iette), is filled with known mount of owder nd shken until no further chnges re observed in the volume occuied by the owder. At this condition the density is considered to be the t density. Higher density owder beds re obtined by comressing the owder inside the tube from both ends. The cylinder is lowered in the beker u to the oint tht the fluid level reches the to of the glss wool. Time zero is chosen when the liquid first meets the owder. The length of infiltrtion nd time re recorded using CCD video cmer connected with n imge nlysis system. Theory The model ssumes orous bed comosed of rndomly cked monosized shericl rticles. Cillry forces drive the infiltrtion rocess. The orous bed structure leds to cillries whose dimensions re relted to rticle size nd bed orosity. The totl ermebility, k cn be exressed in terms of the orosity $\esilon$ nd rticle size D through the Blke-Kozeny eqution (Eq. 1.4). Using Drcy's lw the fluid velocity is given by:

5 dl dt k P (.1) η l where l is the liner extent of fluid in the orous bed, η is the fluid viscosity nd P is the cillry ressure difference. The orous structure of cked bed of monosized shericl rticles of size D cn be ssimilted to n ssemble of cylindricl chnnels with effective rdius determined by the hydrulic rdius theory. For cylindricl tubes, the cillry ressure deends on the liquid surfce tension γ l,v, the owder-fluid contct ngle θ nd the ore rdius r : γ l, v cosθ P (.) r For cked bed, the effective ore size cn be tken s the hydrulic rdius R h, written s D R h (.3) 6(1 ) Substituting Eqs. 1.4,. nd.3 into Eq..1 nd integrting we obtin: l 1 D γ l, v cosθ t (.4) 150 η (1 ) Note tht the hydrulic rdius of cylinder ws considered to be one-hlf of the ore rdius. Eqution.4 is the well known Wshburn eqution. Combining ll mteril rmeters into n infiltrtion rte, W (Wshburn coefficient), Eq..4 cn be simlified to: l Wt (.5) where D 1 W γ l, v cosθ (.6) 150 η (1 ) The Wshburn coefficient deends on owder nd fluid chrcteristics nd fluid-owder interfcil roerties. In deriving the exression for the Wshburn coefficient, we mde the ssumtion of rndomly cked shericl units of dimeter D. In some instnces, D does not necessrily reflect the size of rimry rticles rndomly cked into orous bed. One cn envision lrger units (i.e. smll clusters of rimry rticles) rndomly cked into orous structure. Whether the rimry rticles or lrger units govern the bed ermebility will deend on the cking chrcteristics.

6 Moreover, one lso hs to consider rticle size distribution. Following the led of Dullien nd McDonld [Dullien 1976,McDonld, 1991] we will consider n verge ore size relted to n rent dimeter, D, for the rimry units. Results nd Discussion Infiltrtion exeriments were erformed following two different schemes: cillry rise exeriments with owder beds t t densities nd fluids of vrious viscosities cillry rise exeriments with owder beds t vrious density, using the sme fluid. Figure shows tyicl lot of dt of one infiltrtion exeriment. The liner reltionshi between the squre of infiltrted length, l, nd time t, redicted by Eq..4, cn be observed. The infiltrtion exeriments erformed on silic t t density with different fluid viscosities showed good greement with the theory. In Figure 3 the infiltrtion rtes re lotted versus the recirocl viscosity of the fluids. The liner reltionshi is evident nd from the sloe (lest squre fit method) we cn determine the rent dimeter D, considering surfce tension vlue γ l,v 0.00 J/m ([Krevelen, 1990] nd good wetting of the owder by the olymer (cosθ ~ 1). The dt in Figure 3 indicte vlue of D ~ 700 nm. The vlue is well bove the rimry rticle size clculted from BET surfce dt (8 nm). This indictes very lrge rimry units which my be reresenttive of lrge ggregtes. The sctter observed in these exeriments derive from errors in determining the exct orosity of the smle nd the velocity of the infiltrtion henomen. It is cler tht for fster infiltrtion henomen, one exects lrger sctter. Looking t the exeriments erformed t different densities however, one cn relize tht these lrge ggregtes chrcterize the infiltrtion rocess only t t density. Exeriments t lower orosity were erformed using PDMS of 10 cs viscosity. Results for silic owder re resented in Fig. 4. A lot of Wshburn coefficients versus /(1 ) (which we will cll orosity fctor in this chter) shows liner deendence for high density comcts with shr trnsition t orosity vlues corresonding to the t density (Figure 4). The rent size of the rimry units extrolted from the high density region is 00 nm which is different thn the vlue for D t t density. At t density the size of the ores ers to be much lrger thn t higher comct density, lthough there is only smll difference in the vlue of the overll orosity (e.g, the orosity t t density is 0.88 while high density comcts hve orosity equl or lower thn

7 0.863). The vlue of D extrolted from the high density dt suggests tht the orosity nd, in turn, the ore sizes re determined by voids between smller clusters of rimry rticles, while t t density these seem to be determined by voids between lrger clusters of rimry rticles. For comrison uroses, we undertook the sme exeriments with the other owders. Results of the infiltrtion exeriments erformed with the sme fluid (PDMS, 10 cs) nd owder beds t different densities re resented in Figures 5, 6 nd 7 for CCO 3,, TiO $ nd crbon blck resectively. The rent size of structurl units determined for the three owders t high density re reorted in Tble. Clcium crbonte nd titnium dioxide hve n rent rticle size which is close to the rimry rticle size. This suggests tht rimry rticles in this cse well reresents the rimry units which govern infiltrtion in this density rnge. Crbon blck shows size lrger thn rimry rticles indicting, s in the cse of silic, tht infiltrtion my be ffected by smll clusters of rimry rticles. However, both clcium crbonte nd titnium dioxide still show the sme trnsition for orosity close to the t density, but crbon blck did not show the sme tye of trnsition. The lrge difference in the rent dimeter of rimry units in owder beds t t density (D TAP ) nd the vlues obtined with comcted structure (D HIGH ) suggests tht imortnt structure rerrngements hen uon owder comction. Powder Comction Model To exlin the strong deendence of the ermebility uon comction density, we ssume hierrchicl structure nd consider two levels: rimry nd secondry units. In most cses these cn be identified with rimry rticles nd ggregtes. It is logicl to ssume tht the inter-ggregte ores hve lrger size thn the intr-ggregte ores. However the rocess of comction my chnge the role tht inter-ggregte nd intr-ggregte orosity ly in the infiltrtion rocess. Adms [Adms, 1994] describes the rocess of comction dividing it into two serte stges. During the first stge owder is comressed strting from its t density. In this stge lrge conformtionl rerrngements occur nd high ressure is not needed to comress the owder. In the second stge filure of gglomertes nd interenetrtion of ggregtes occurs nd the ressure needed for comction increses exonentilly with the strin of the comct. A very similr

8 exlntion is offered lso by Song nd Evns [Song, 1994]. It is during the first stge nd the beginning of the second tht we believe the lrgest chnges in ore size occur. To the hierrchicl structure of the gglomertes, we cn ssocite hierrchicl structure of the ores. Generlly seking, voids between ggregtes re lrger thn voids within ggregtes (lthough it deends on the cking stge). At t density the concentrtion of lrge ores my be enough to crete comlete th of lrge ores for the fluid (ercoltion threshold). Flow through lrger chnnels will be fster thn in the smller ores. In the exeriments we would be ble to detect only the fstest flow flux in the orous medi. Infiltrtion within ggregtes will occur with slower kinetics nd we would not be ble to observe it directly. When the owder bed is comcted nd rerrngements occur, the ermebility chnges. As the concentrtion of lrge ores decreses the lrge th for fluid becomes blocked nd fluid cn trvel only through the smll ths. An dditionl increse in the density cuses interenetrtion of the ggregtes reducing their inner orosity. At this level, the owder bed cn be ictured s collection of monodisersed chnnels. This henomenologicl icture cn be formulted in mthemticl fshion nd exlin the exerimentl results. The Blke Kozeny eqution nd its modifictions due to Dullien nd McDonld tke into ccount collection of rticles of single or different size, but they do not consider hierrchicl structure for the owder bed. A orous bed hs totl volume V T, which is the summtion of the ggregte volume, V, nd the void volume mong them, V v. Therefore we cn write: V V + V (4.1) T v The volume of the ggregtes cn be in turn decomosed in rimry units volume V nd void volume V v V V + V v ' (4.) We define inter-ggregte orosity the rtio of the void volume between ggregtes, V v, nd the totl volume V T : from which follows tht V v (4.3) VT

9 V ( 1 ) V T (4.4) Similrly we define the intr-ggregte orosity,, s the rtio of the void volume mong rimry units nd the totl volume of the ggregte V ' v V (4.5) The volume occuied by the rimry units cn be conversely written in this fshion: V ( 1 ) V (1 )(1 ) V (4.6) T Here is the orosity of the owder bed considering solid ggregtes, while is the orosity of single ggregte. The totl void volume of the owder bed cn be obtined summing the void volume between nd within ggregtes. Combining Eq we obtin the exression for the totl orosity (V v +Vv )/V T : + 1 ) (4.7) ( Eq. 4.7 exresses the contribution of the inter-ggregte nd intr-ggregte orosity to the overll orosity. In the cse of very loose gglomertes ( O(1)), the inter-ggregte orosity is the dominnt term, while in cse of close cked ggregtes ( ~ 0) the orosity is determined by the intr-ggregte orosity. In the cse of rndomly cked shericl rimry units, where no hierrchicl structure cn be identified ( 0), Eq. 4.7 reduces to n identity. We cn clculte the hydrulic rdius for intr-ggregte ores nd for inter-ggregte ores s follows: R h cross section vilble for flow wetted erimeter intr ggregte volume vilble for flow totl wetted rimry units surfce ρ S u V ' v (1 ) V D 6(1 ) (4.8) with S u the secific surfce of the rimry units, ρ their density nd D their size. The two quntities re relted through the following eqution:

10 D 6 (4.9) ρ S u Similrly we cn derive the hydrulic rdius for the inter-ggregte orosity, considering solid ggregtes nd noting tht in this cse the wetted surfce is only the externl surfce of ggregtes: R h cross section vilble for flow wetted erimeter inter ggregte volume vilble for flow totl wetted ggregte surfce Vv ρ S (1 ) V T D 6(1 ) (4.10) where S is the secific surfce of the ggregtes, ρ their density nd D their size. The velocity of lminr flow in circulr tube of rdius R for fluid with viscosity η is [Bird, 1960]: R dp r v z 1 ( ) 4 (4.11) η dz R The suerficil velocity is defined s v0 v where v is: v 1 πr h Rh 0 π 0 v rdrdθ z PR 8η l h (4.1) where we hve relced the rdius R with the hydrulic rdius R h nd P cn be exressed s in Eq... In the resence of hierrchicl structure of ores with two min sizes, the medi will be infiltrted by two flows with different flow rtes. The suerficil flow velocity (v 0 ) between ggregtes of men size D cn be written in this fshion: v γ cosθ D Similrly one cn derive the velocity within one ggregte v 0 : l, v 0 (4.13) 48η l 1 γ l, v cosθ D v0 (4.14) 48η l 1

11 where l nd l is the level of fluid in the orous medi between nd inside ggregtes resectively. The flow velocities clculted for the two serte chnnels deend on chrcteristic size, ggregte nd rimry rticle size resectively. A first comrison my indicte tht the suerficil velocity between ggregtes my be lrger due to the lrger chrcteristic size (D >>D ) thn the velocity inside the ggregtes. Without being ble to mesure the degree of sturtion of the infiltrtion rocess (s in the Wshburn exeriment) one would detect the fstest flow nd therefore the different regimes of infiltrtion uon comction. However full comrison of the velocities hs to tke into ccount the reltive contributions of inter nd intr-ggregte orosity s function of totl cking density. In the hysicl icture given bove we serted the comction rocess into two stes: first rerrngement of ggregtes without interenetrtion followed by n interenetrtion of the ggregtes which leds to chnges lso in the intr-ggregte orosity. We consider s initil stge the t density which will corresond to n initil totl orosity i. We cll i nd i the initil intr-ggregte orosity nd inter-ggregte orosity. At t density we ssume tht ggregtes re not interenetrted nd i cn be identified with the orosity of the ntive ggregtes. Therefore we cn derive the initil inter-ggregte orosity from the initil overll orosity i. through Eq. 4.7: i i i (4.15) 1 i The first stge of rerrngement of the ggregtes is quntified by reduction in the interggregte orosity ( ) with n unchnged intr-ggregte orosity ( i ) nd it lsts until ggregtes hve reched mximum cking configurtion (or minimum orosity ) nd cnnot be rerrnged nd cked ny further without interenetrtion. Incresing the density cuses ggregtes to interenetrte nd reduce the intr-ggregte orosity. As interenetrtion occurs, ores between ggregtes re soon filled nd the owder comct cn be seen s collection of rimry rticles ( 0). We ssume vlue of orosity (criticl orosity ) which reresents the stge where sheres cnnot rerrnge further without interenetrtion. The criticl orosity therefore reresents the

12 trnsition between the two stges exlined bove: first rerrngement of ggregtes ( > ) followed by rerrngement nd interenetrtion of ggregtes ( < ). Figure 8 resents henomenologicl icture of the stges of the comction rocess described bove. As consequence of the comction rocess one cn strt to icture the reltive contributions of orosity between nd inside ggregtes, reresented in Fig. 8 s white nd blck ttern chnnels resectively. Chnnels between ggregtes (white) re resent t low density nd their contribution is drsticlly reduced s rerrngement occurs. Model Predictions The ctul reltive contributions of inter nd intr-ggregte orosity deend on morhologicl roerties of the owder, such s the density of the ntive ggregtes ( i ) nd the bility of ggregtes to rerrnge nd ck ( ). We ssume, s first roximtion, to hve monodisersed shericl ggregtes ble to ck in rndom fshion. Therefore we fix to 0.36 (orosity of collection of rndomly cked monodisersed sheres [Zllen, 1983] nd nlyze the two contributions nd s function of the owder bed cking density nd the density of the ntive ggregtes. The endix section resents in more detil the ssumtions used to clculte the reltive contributions of nd s function of cking density. Tking into ccount Eq. 4.7, we cn clculte the intr-ggregte nd inter-ggregte orosities s function of the owder bed solid volume frction for two cses, one considering very loose ggregtes ( i 90%) nd one considering dense ggregtes ( i 70%). Results re shown in Figures 9 nd 10 resectively, where we lotted the vlues of inter-ggregte orosity nd intrggregte orosity s functions of cking density. For very loose ggregtes ( i 90%) one cn observe tht the contribution from inter-ggregte orosity is lwys lower thn the intr-ggregte orosity. For dense ggregte insted, chnge in the trend is exected. At high density the intr-ggregte orosity is leding, while t lower density the inter-ggregte orosity gives the higher contribution. At lower density, combining the greter extent of inter-ggregte ores with their lrger size, the flow mong ggregtes cn be dominnt. However, uon comction when interenetrtion occurs ( ) the contribution of the inter-ggregte ores decreses. At this oint, the fstest flow will be within the ggregtes,

13 suggesting tht chnge in the regime of the infiltrtion rocess my occur. Therefore two min regimes could govern infiltrtion: one chrcterized by lrge intr-ggregte ores (D ) nd the second through inter-ggregte ores of smller size (D ). The trnsition shown in the exeriments cn reflect the chnge in the infiltrtion regime uon comction. So fr, the two orosities nd hve been comred nd their contribution to ermebility ssessed in terms of their reltive mgnitude. However in the cse of inter-ggregte orosity, one hs to tke into ccount tht the inter-ggregte ores hve lrger volume, leding to lrger contribution to the overll ermebility even when the ercentge is low. To hve better comrison of the two contributions one hs to know D nd the reduction in size of the inter-ggregte ores when interenetrtion occurs. This is however extremely difficult to redict. A simlifiction is to consider the ggregtes s collections of shericl rimry rticles nd to derive single hydrulic rdius. This entils tht while fluid is trveling through smller or lrger ores (intr-ggregte nd inter-ggregte ores), it is lwys wetting the entire oen surfce. Thus we clculte R h s the rtio between the totl ore volume vilble for fluid infiltrtion nd the totl wetted surfce: R h V ρ S cross section vilble for flow wetted erimeter volume vilble for flow totl wetted surfce v u + V ' v (1 ) V T VT + (1 ) VT ρ S (1 ) V D D + 6(1 )(1 ) 6(1 ) u T (5.1) R h still shows two min contributions belonging to the different orosities. Following the sme line used to derive Eq nd 4.14, we cn write the suerficil velocity in this fshion: γ l, v cosθ DP DP v0 + DP ( 1 ) ( )( ) ( ) + ( (5.) 8η l ) The Wshburn coefficient will therefore be: W γ l, v cosθ η D P ( 1 )( 1 ) + D P D 6 1 P ( 1 ) ( ) + ( ) 1 (5.3)

14 The sme result cn be obtined by relcing in Eq..6 with its exression from Eq Looking closer to Eq. 5.3 we cn recognize minly three terms which contribute to the infiltrtion D rte. The first term ( ) reresents the inter-ggregte orosity nd it hs size ( ) lrger thn the rimry rticle size. However the size will chnge s soon s strts to chnge, D which hens when interenetrtion occurs. The second term ( D (1 ) 1 ) reresents the effective orosity within the ggregtes while the third term ( D 1 ) tkes into ccount the collse of lrger ores into smller ones when the overll orosity is reduced. The second nd third term my be groued together nd Eq. 5.3 rewritten in different fshion: W k k γ l, v cosθ D + D 150 ηdp ( 1 )( 1 ) ( 1 ) ( ) + ( ) (5.4) In this wy the hysicl icture is more cler. The first term in Eq. 5.4, k, reresents n interggregte ggregte ermebility while the second term, k, reresents the intr-ggregte ermebility. For this ltter term there re two min contributions: the inner orosity of the ggregtes nd the collsed ores between ggregtes. Eq. 5.4, derived by ssuming tht the fluid is wetting the entire oen surfce (see Eq. \ref{hydrrdius}), is rorite for the high density region. At low density, the k term redicts chnge in the infiltrtion regimes, but it my not be ble to fully reresent the fst infiltrtion rte observed t TAP density, where it is more resonble to believe tht the fluid will wet only the ggregte surfce. We cn now comre the contributions which ech term gives to ermebility s function of totl orosity. Following the sme line of the revious comrison we clculted the two contributions k nd k of Eq. 5.3 normlized on the D term nd lotted them versus the orosity fctor /(1-). Results re resented in Figures 11 nd 1. In the cse of loose ggregtes ( i 90%) k domintes Eq. 5.3 similrly to the revious comrison, while in the cse of dense ggregtes ( i 70%) there is chnge in the dominnt term. The inter-ggregte orosity domintes t low density while the

15 intr-ggregte orosity domintes t high density. The trnsition however hens t lower vlue of totl orosity thn illustrted in Fig. 10. This is the effect of the lrger size of the interggregte ores. It is imortnt to note tht the trnsition between the two regimes occurs before reches the criticl vlue (0.36). It is indeed the strong reduction of lrger ores uon ggregte reorgniztion tht drsticlly decreses the contribution of the lrger ores. However the lrger size of the inter-ggregte ores mke the inter-ggregte flow still dominnt even if the reltive contribution is lower thn the intr-ggregte contribution. The criticl orosity 0.36 ws bsed on the ssumtion of monodiserse shere rndomly cked. However, ggregtes my hve different sizes nd shes, which cn led to interenetrtion t higher vlues of inter-ggregte orosity. Although this could chnge the reltive contributions of nd s function of cking density, the generl results still hold true. A direct comrison of the model with exerimentl results requires morhologicl informtion of the owder, in rticulr the vlue of orosity for the ntive ggregtes. Unfortuntely mong the owders tested, this vlue is known only for crbon blck. In this cse i cn be clculted from the dibutyl hthlte (DBP) bsortion vlues (in units of cm 3 /100g) through the following eqution [Ymd, 1997]: DBP (1 ) (5.5) The initil orosity (t orosity) cn be mesured exerimentlly nd through Eq. 5.5 we cn clculte the initil intr-ggregte orosity i. For the crbon blck tested (Monrch 880) the t orosity is 0.93 nd initil intr-ggregte orosity is This indictes morhology of very dense ggregtes. In Figure 13,, nd re lotted versus the orosity fctor /(1-) for the initil vlue of clculted. The intr-ggregte orosity ers to be lwys smller thn, exlining the single regime in the infiltrtion rocess s exhibited by crbon blck. This is concurrent lso with the size D determined from the exerimentl results, which is more reresenttive of ggregte

16 size rther thn rimry rticles. In the rnge of densities tested, the morhology of this owder cn be esily reresented s collection of very dense ggregtes. For the other owders tested in this work, the intr-ggregte initil orosity is not known. However, one my seculte tht trnsition between the two infiltrtion regimes occurs when the contributions from the inter-ggregte nd intr-ggregte orosities on the overll ermebility become comrble. This ssumtion (k ~ k ) trnsltes into: t 1+ (5.6) 1+ which uon relcing in terms of the overll orosity nd (from Eq. 4.7) gives 1+ i t i (5.7) 1+ i i In Eq. 5.7 we hve considered t the totl orosity t the trnsition regime where (the intrggregte orosity) hs not been ffected by the comction rocess nd therefore cn be relced by the ntive ggregte orosity i. Exerimentl results showed chnge in the infiltrtion regime for vlues of /(1-) between 5 nd 7 (Figures 4-6) nd they were use to estimte the ntive intrggregte orosity i for silic, CCO 3 nd TiO. The vlues, summrized in Tble 3, show highly orous structures for the ggregtes of these owders A closer look t Eq. 5.3 revels nother interrettion of the infiltrtion results. At smll overll orosity, owder bed ermebility will be governed by the term reresenting the intr-ggregte orosity, k, while t lrger orosities it is the flow between the ggregtes which domintes. In other words, t vlue of below t, owder bed ermebility is well roximted by the intrggregte orosity contribution k,. At vlues of bove t, the dominnt contribution is relted to inter-ggregte orosity, i.e. k. Indeed one cn clculte the contributions to the overll system ermebility where the trnsition oint t is relted to the ntive ggregte density i nd lot the dominnt contributions ginst the exerimentl results. Figures show these lots for the vrious systems investigted together with the exerimentl results. In fitting the exerimentl dt with the model we hve extrolted D from the sloe of the Wshburn coefficient versus the intr-ggregte ermebility contribution k. It is evident in the cse of silic nd CCO 3 tht the fittings show

17 ositive intercet with the y xis. This is most likely due to the fct tht the Blke-Kozeny formultion is not ble to describe ermebility of owder t very low orosity nd tht closed orosity is robbly resent in the owder bed. By fitting the dt t low orosity where the liner behvior is observed, one cn ctully exlin the chnge in the infiltrtion regime by the incresing contribution of the inter-ggregte orosity to ermebility. The infiltrtion rtes observed in the high orosity region re even lrger thn the ones redicted by the model. When we serted the two contributions to the hydrulic rdius theory for interggregte nd intr-ggregte orosity, we mde the ssumtion tht the fluid between ggregtes would wet the entire surfce of the rimry rticles. From henomenologicl oint of view, one cn envision tht the wetted surfce by the fluid would be smller nd smller s the orosity increses, reching eventully the stge where only the externl surfce of ggregtes is wetted. This would trnslte in fster flow between ggregtes which cn esily exlin the exerimentl results. Conclusions The Blke-Kozeny eqution ws modified tking into ccount hierrchicl structure of the owder bed, considering different cking behvior t the level of ggregtes nd rimry rticles. We distinguished between inter-ggregte orosity nd intr-ggregte orosity nd modified the Blke-Kozeny eqution highlighting the contributions tht both orosities give. At high orosity the inter-ggregte ores dominte in the infiltrtion rocess, while, t higher density, the intr-ggregte terms become the most imortnt. This cn exlin sudden chnge observed in the exeriments of infiltrtion kinetics tht occurs uon comction of the owder bed bove its t density. Acknowledgment The finncil suort from Pirelli Pneumtici S.. A. is grtefully cknowledged.

18 Aendix Assuming shericl ggregtes, one cn clculte the extent t which interenetrtion will fill the inter-ggregte ores. Although the secific vlues deend of course on the configurtion of the ggregtes, however the differences mong different configurtions re robbly irrelevnt for the urose of this study. We considered, s n exmle, n ssemblge of shericl ggregtes in n octhedrl configurtion. This is reresented s lnes of shere, with ech shere on one lne t the center of four djcent sheres in the lne below nd bove. The schemtics is resented in Fig. 17. Assuming unixil comression, the ores between the sheres will be filled when the center of the sheres of the uer lne will be tngent to the to surfce of the sheres of the lower lne. This llows us to clculte the chnges in void frction inside the sheres in order to relte the chnges in with. Ech uer shere increses its density due to the intersection with the shericl cs of four sheres below. The intersection volume cn be clculted indirectly considering cube of side r with ech vertex in the center of one shere. The intersection volume between the cube nd shere is one eighth of the volume of the shere which leds to void volume: V void r 1 (A.1) 6 3 π When the void is filled, the intersection volume between the two sheres is the volume of the cube minus twice the void volume. The totl intersection volume of the shere with the lower lne is therefore: π δv 4 3 int r 1 (A.) 3 The solid volume frction of shere A will therefore chnge from its vlue φ to vlue φ': φvsh + φv ' int φ φ + δφ V sh (A.3) where V sh is the volume of the shere nd δφ reresents the dded solid volume frction φδv int /V sh.

19 The chnges in orosities cn be esily clculted nd led to the following exression: ' ' δv int δv int 1 φ i δφ i 1 + (A.4) V sh Vsh where i is the void frction of n unmodified shere. Thus there is liner reltionshi between the chnges in void frction inside nd between the sheres. The vlue of δv int is which leds to the conclusion tht smll chnges in the void frction of the sheres is enough to fill comletely the ores between them. The numericl vlue is relted to the octhedrl configurtion we hve nlyzed. However one cn generlize nd extend the clcultion to different configurtions. These ssumtions cn be trnslted in mthemticl formultion: * i i ' * ' min( i, ) ' i o ' 0 0 (A.5) The first stge is resent only if the initil inter-ggregte orosity is lrger thn the criticl orosity. If i < it is not meningful to consider hierrchicl structure. A low inter-ggregte initil orosity mens tht we do not recognize ggregtes s serte structure governing infiltrtion. The orosity i, in this cse, would reresent lrger ores due to loose rrngement of lrge clusters of rimry rticles. The second term in Eq. A.5 describes the collsing of lrge ores into smll ones. The rocess is fst comred to the reduction of the overll orosity. This should be reresenttive of the fct tht interenetrtion cuses lmost immedite filling of the lrge ores between ggregtes. The lst stge reresents reduction in orosity when no ggregtes re ny longer resent. References

20 [Adms, 1994] Adms, M. J., Mullier, M. A. nd Seville, J. P. K., ``Agglomerte strength mesurement using unixil confined comression test'', Powder Technology, 78, 5-13 (1994) [Alchemy Progrm]{lchemy} Alchemy softwre ckge by Scivision [Bird, 1960] Bird, R. B., Stewrt, W. E. nd Lighfoot, E. N., ``Trnsort Phenomen'', John Wiley & Sons, New York (1960) [Bohin, 1994]} Bohin, F., Feke, D. L. nd Mns-Zloczower, I., ``Penetrtion of silicone olymers into silic gglomertes nd its influence on disersion mechnism'', Rubber Chem. Techn., 67, 60 (1994) [Brunuer, 1940] Brunuer, S., Deming, L. S., Deming, W. E., nd Teller, E., ``A theory of the vn der Wls dsortion of gses'', J. Am. Chem. Soc., 6, (1940) [Dullien, 1976] Dullien, F. A. L., El-Syed, M. S. nd Btr, V. K. ``Rte of cillry rise in orous medi with nonuniform ores'', Journl of Colloids nd Interfce Science, 60, (1976) [Krevelen, 1990]{krevelen90} Vn Krevelen, D. W. ``Proerties of Polymers: Their correltion with chemicl structure; Their numericl estimtion nd rediction from dditive grou contribution'', Elsevier, New York (1990) [McDonld, 1991] McDonld, M. J., Cho~-Feng, C., Guilloit P. P., Ng, K. M., ``A generlized Blke~-Kozeny eqution for Multisized Shericl Prticles'', AIChE Journl, 37, 1583 (1991) [Ottino, 000] Ottino, J. M., ``Mixing nd disersion of viscous liquids nd owdered solids'', Advnces in Chemicl Engineering, 5, (000) [Schediegger,1957] Adrin E. Schediegger, The Physics of flow through orous medi, The Mcmilln Comny, New York (1957) [Song, 1994]{song94} Song, J. H. nd Evns, J.R.G., ``A die ressing test for the estimtion of gglomerte strength'', J. Am. Chem. Soc, 77, (1994) [Wshburn, 191] Wshburn, E. W. ``The dynmics of cillry flow'', The Physicl Review, 17, 73 (191) [Ymd, 1997] Ymd, H. ``The influence of infiltrtion on the disersibility of crbon blck gglomertes'', PhD Thesis, Cse Western Reserve University (1997)

21 [Zllen, 1973] Zllen, R., The Physics of Amorhous Solids, John Wiley & Sons, New York (1973)

22 TABLES Powder ρ (g/cm 3 ) S BET (m /g) DBPA D rim (nm) (cm 3 /100g) SiO. 98 NA 8 CCO NA 50 TiO NA 167 Crbon Blck Liquids 5 C (cs) PDMS 10, 50, 100, 500, 1000 Tble 1:Min chrcteristics of owders nd olymers used in the exeriments. ρ indictes the density of rimry rticles, S BET indicted the BET surfce, D rim indictes the rimry rticle dimeter clculted from the BET surfce [Bird, 1960]. DBPA is the dibutyl htlte bsortion number.

23 Powder D rim (nm) TAP D (nm) (t density) D (nm) (high density) SiO CCO TiO Crbon Blck Tble : Structurl chrcteristics for selected owders extrolted from infiltrtion exeriments. TAP indictes the totl orosity of the owder bed t t density

24 Powder t t SiO CCO TiO Tble 3: Vlues of intr-ggregte orosity for different owder extrolted from the exerimentl results.

25 Figure 1: Schemtic reresenttion of the exerimentl rtus for mesurement of infiltrtion. A cylindricl glss is filled with owder nd lowered into beker full of liquid of known viscosity. The length of infiltrted owder is recorded s function of time.

26 Figure : Rw dt for Wshburn exeriment. The liner reltion between l nd time t is evident.

27 Figure 3: Plot of Wshburn coefficients for fluids of different viscosities η infiltrting silic owder t t density.

28 Figure 4: Plot of Wshburn coefficients t different orosities for infiltrtion of PDMS into silic owder.

29 Figure 5: Plot of Wshburn coefficients versus orosity for CCO 3.

30 Figure 6: Plot of Wshburn coefficients versus orosity for TiO.

31 Figure 7: Plot of Wshburn coefficients versus orosity for crbon blck owder.

32 Figure 8: Schemtic reresenttion of the comction rocess.

33 Figure 9: Plots of nd versus owder bed solid volume frction for very loose ggregte ( i 0.9).

34 Figure 10: Plots of nd versus owder bed solid volume frction for dense ggregtes ( i 0.7).

35 Figure 11: Plot of the two min contributions k nd k to the ermebility for initil intr-ggregte orosity of 90%

36 Figure 1: Plot of the two min contributions k nd k to the ermebility for initil intr-ggregte orosity of 70%

37 Figure 13: Plot of, nd versus /(1 ) for owder with initil intr-ggregte orosity 0.57, s clculted for Crbon Blck.

38 Figure 14: Plot of the Wshburn coefficient for Crbon Blck. i 0.57 (from DBPA) which corresonds to t 0.8. Also shown re the exerimentl results.

39 Figure 15: Plot of the Wshburn coefficient for CCO 3. t 0.85 which corresonds to i 0.64 Also shown re infiltrtion exeriments results.

40 Figure 16: Fit of the inter-ggregte nd intr-ggregte contributions to the infiltrtion rte for silic owder.

41 Figure 17: ) Reresenttion of octhedrl configurtion; side view nd b) to view. c) Configurtion corresonding to 0.