Effect of the Hydophobc Foce Stength on Patcle- Bubble Collson Knetcs: A DEM Appoach Ya Gao 1, Geoffey M. Evans 1, Eca J. Wanless 2 and Robeto Moeno-Atanaso 1 1 School of Engneeng 2 School of Envonmental and Lfe Scences The Unvesty of Newcastle, Callaghan, NSW 2308, Austala Emal: Robeto.Moeno-Atanaso@newcastle.edu.au Abstact The captue of sold patcles by a bubbles s a complex pocess nfluenced by hydodynamc and suface foces. Compute smulaton povdes an altenatve way to expemental appoaches to gve an nsght nto the phenomenon of patcle captue. In ths pape, the theedmensonal Dscete Element Method (DEM) has been appled to smulate the nteactons between fne patcles and a cental bubble n a quescent lqud. A system, consstng of 200 monodspese slca patcles and a statonay a bubble, was consdeed. Dag, buoyancy, hydodynamc esstance, hydophobc and gavtatonal foces have been ncluded n the smulatons. The attactve hydophobc foces between the patcles and the bubble was estmated though a sngle exponental decay law whch depends on two paametes, K and λ. K s elated to the maxmum stength of the hydophobc foce, whle λ s the length that ndcates how fast the hydophobc foce decays wth patcle-bubble dstance. The esults have shown that a decease n the ato of maxmum hydophobc foce to patcle weght fom 5.2 10 6 to 5.2 poduced a decease n the captue effcency by 42.6% fo λ=1 μm and by 12.5% fo λ=1 nm. Keywods- Dscete Element Method; flotaton; hydophobc foce I. INTRODUCTION Foth flotaton s an mpotant ndustal pocess extensvely used n the sepaaton of mneal patcles as well as n the teatment of wastewate. Flotaton s geneally descbed as a sequence of thee sub-pocesses, namely, collson, attachment and detachment [1-5]. The mpovement of the pefomance of the flotaton pocess geatly depends on povdng adequate undestandng of the vaous sub-pocesses and detaled knowledge of the nteacton foces between patcles and bubbles. Theefoe, study of nteactons between sold patcles and a bubbles n aqueous solutons has been the focus of an nceasng amount of eseach n undestandng foth flotaton. Howeve, due to the complexty of the flotaton phenomenon, the pncples govenng the bubble-patcle nteactons ae not well undestood despte many decades of eseach. The nteactng foces between hydophobc collodal patcles have been well explaned by the classcal DLVO theoy [6, 7]. Howeve, when studyng the nteacton between mneal patcles and a bubbles the stuaton gets much moe complcated due to the pesence of non-dlvo suface foces, such as stec foces, hydaton and hydophobc foces [8, 9]. Undoubtedly, the hydophobc foce s consdeed as the most sgnfcant non-dlvo suface foce that detemnes the captue of a patcle by a sng bubble dung flotaton [10]. Hydophobc nteactons, on a molecula level, descbe the elatonshps between wate and hydophobes whch ae nonpola molecules that do not nteact favoably wth wate molecules. Ths foce s thought to appea due to the coalescence between nonpola molecules dstbuted on appoachng sufaces, because, n ths way, the contact wth wate molecules can be educed to lowe the fee enegy [11]. The hydophobc foce has geneally been found to be an attactve foce that can ncease wth the macoscopc hydophobcty of sufaces, and to be fa stonge than the van de Waals attacton between sufaces [12]. Howeve, the ogn of the hydophobc attacton s complcated and s stll contovesal at pesent. Isaelachvl and Pashley (1982) [13] demonstated that the nteacton between hydophobc sold sufaces was about ten tmes stonge than the maxmum possble van de Waals foce. Afte the poneeng wok by these authos, a lage amount of studes have focused on the detemnaton of the stength of the hydophobc foce by usng expemental technques, such as suface foces appaatus (SFA) and atomc foce mcoscope (AFM). Table 1 pesents a bef evew of ecent pogess n undestandng the hydophobc foce n aqueous meda. Despte the lage effots to detemne the stength of the hydophobc nteacton, thee s stll no geneally accepted mathematcal expesson of t. Ths s due to a numbe of factos such as: Most of the epoted expemental data ae wthn the ange of hundeds of nanometes between the nteactng sufaces. Ths s due to the wokng ange
TABLE I. BRIEF REVIEW OF HYDROPHOBIC FORCE Autho(s) Hydophobc Foce, Technque Notes and Symbol Defntons Isaelachvl and Pashley (1982) [13] Sngle exponental decay law: / R = K exp( H / λ) Suface Foces Appaatus (SFA) Hydophobc sufaces: hexadecyltmethylammonum bomde monolaye-coated mca sufaces. Long-ange foce can act up to 10 nm. R s the adus of the cuved sufaces, K s a constant, H s the sepaaton dstance and λ s the decay length ( 1 nm). Claesson et al. (1986) [17] / R = K exp( H / λ) SFA Hydophobc sufaces: doctadecyldmethlammonum bomde monolaye-coated mca sufaces. Measuable ange: about 30 nm. Claesson and Chstenson (1988) [18] Double exponental decay law: / R = K exp( H / λ) + K * * exp( H / λ ) SFA Hydophobc sufaces: unchaged hydocabon and fluoocabon monolaye-coated mca sufaces. Vey long-ange foce has measuable ange aound 80 nm. λ=2-3 nm; λ * =13 nm (hydocabon) and16 nm (fluoocabon). Yoon et al.(1997) [14] Powe law: 2 / R = K / H cosθ + cosθ2 K = exp( a + 2 1 b ) Atomc Foce Mcoscope (AFM) Hydophobc nteacton between dssmla sufaces: octadecyltchlooslane (OTS) coated glass sphee and slca plate sufaces. λ =2-32 nm. θ 1, θ 2 ae contact angles of two sufaces. a=-7.0, b=-18.0 Ducke et al.(1994) [15] AFM Hydophobc sufaces: a bubble suface and OTS monolaye-coated slca patcle suface. Long-ange attactve foce exsts between hydophobc patcle and bubble. Ishda and Hgashtan (2006) [16] AFM Hydophobc sufaces: slca patcle suface coated wth OTS. A stong long-ange attacton between hydophobc sufaces wth nanobubbles (not efeed as hydophobc attacton ). A shot-ange attacton foce between hydophobc sufaces wthout nanobubbles (genune hydophobc attacton). of AFM measuement technque. Exstence of the hydophobc foce beyond these dstances has not been exploed yet [14-16]. Patcle manpulaton po to expements (hydophobzng and dyng) may ntoduce nanobubbles on to the patcle sufaces, whch may affect the valdty of the analyss [19]. Bubble defomaton due to the appoachng patcle befoe the fomaton of the thee-phase contact (TPC) lne also poses a poblem that can lead to ncoect esults [4]. Although many eseaches [18, 20-22] have made sgnfcant contbutons to addess these poblems, thee ae stll many aspects of the hydophobc nteacton that ae not well undestood. These aspects nclude ts ogn, ange and stength descpton. The use of computatonal methods n the analyss of patcle-bubble systems has eceved nceasng attenton n ecent yeas. Dscete Element Method (DEM), as one of numecal methods n patcle technology, seves as a poweful tool fo nvestgatng the detaled phenomena n the patclebubble nteacton as well as descbng the nteacton foces of dffeent knds. In such a way, the DEM model allows us to pobe the ntenal state of the system and undestand the fundamental patcle-bubble nteactons undelyng the complex, maco-scale esponse. In 2012, Maxwell et al. [23] developed a DEM code wtten n Fotan 77 fo studyng the patcle-bubble nteactons n monodspese and polydspese systems togethe wth the patcle sldng tme ove the bubble suface. In the wok, the hydophobc nteacton was assumed to follow a elatonshp wth the nvese of the patcle-bubble suface dstance. Aftewads, Moeno-Atanaso [24] analysed the effects of thee dffeent models fo
hydophobc foce epoted n the lteatue on the patcle captue effcency. The wok pesented hee bulds upon the pevous fndngs by Maxwell et al. (2012) and Moeno-Atanaso (2013). The objectve of the pesent pape s to nvestgate the nfluence of the stength of the hydophobc foce on the knetcs of captue of patcles by the bubble. The hydophobc attacton s descbed by a sngle exponental law whch takes patcle hydophobcty and decay length nto account. Addtonally, the buoyancy foce s ntoduced n the smulaton whch was excluded n the studes mentoned above. Van de Waals, electcal double-laye and hydophobc foces between patcles have not been ncopoated nto the DEM smulatons. Ths s due to the fact that these foces wll ntefee wth the elucdaton of the nfluence of the patclebubble hydophobc foce on the captue effcency. Howeve, the ole of the patcle-patcle nteactons on patcle captue by a cental bubble wll be addessed n futue. II. METHODOLOGY A. Geneal Model Descpton The most mpotant featue of dscete element models (DEMs) s that all patcles ae explctly consdeed as ndvdual bodes n the nteactons and movements. The soft sphee model [25], whch allows mno defomatons between contactng bodes, s adopted n the smulatons pesented hee. Ths model was fst poposed by Cundall and Stack (1979) fo quas-statc defomaton of a patcle bed [26]. The contact foces ae smply computed usng lnea elastc spng model n the nomal decton. Elastc popetes wee also assgned to the bubble as descbed n the wok by Attad and Mklavcc, o Goldman [27, 28]. Fo the DEM smulatons, compute code n C language was developed. A system made of 200 monodspese slca patcles and an a bubble was employed fo the numecal computaton. The ntal state of the system s shown n Fg. 1. The bubble wth mmoble suface was fxed to be statonay at the cente of the wokng space thoughout the smulaton pocess. The pmay patcles wee andomly geneated aound the cental bubble wthn a maxmum dstance fom the bubble suface equal to 50μm. These patcles wee constaned to not ovelap wth each othe o wth the bubble at the begnnng of the smulatons. All patcles and the bubble wee assumed to be sphecal n shape. The patcle veloctes wee ntally set to zeo. The popetes of wate wee consdeed to descbe the flud phase. The computaton was pefomed on the bass of quescent flow of the lqud (wate). So the analyss of the nfluence of the elatve stength of the hydophobc nteactons can be dectly studed. B. Equaton of Patcle Moton and Numecal Integaton Scheme The moton of the patcles s descbed by the equaton of Newton s second law. Fo the th patcle, the moton equaton can be wtten as m a = F. (1) whee a s the acceleaton of the cente of the patcle ; m, ts mass; F, the esultant foce. The esultant foce, n the pesent wok, conssts of the long-ange foces (gavtatonal, hydophobc foce), hydodynamc foces (dag, buoyancy, esstant foce), and the elastc contact foce. In the dscete element method, the patcle and patclebubble nteactons ae calculated cyclcally and the evoluton of the system s detemned usng a tme dependent explct soluton method. The half-step leapfog Velet (LFV) ntegaton scheme s used n the smulaton due to ts hgh accuacy (mathematcally), stablty and computatonal effcency [29]. In ths scheme, the velocty s calculated at each half tme step, whch s an mpotant modfcaton based on the Velet algothm. The tanslatonal moton at the nth tme step s gven by: n F a n =, (2) m n+ 1/ 2 n 1 / 2 n v = v + a t,. (3) n+ 1 n n+ 1 / 2 x = x + v t,. (4) n+ 1 n+ 1 / 2 1 n v = v + a t.. (5) 2 whee v s the patcle velocty and x the dsplacement. The soluton poceeds n small tme steps Δt dung whch the acceleaton of the patcles s assumed to be constant. Evey tme step s stated wth computng the patcle acceleatons fom Eq. (2), followed by pefomng the numecal ntegaton of acceleatons ove the tme ncement to obtan the patcle veloctes at next half tme step (Eq. (3)). Fgue 1. Vsualsaton of the ntal state of the system. (Patcles ae coloued n yellow and the bubble n blue.)
Futhe, the dsplacements ae calculated usng Eq. (4) afte whch the velocty at the next full tme step s obtaned by Eq. (5). Havng detemned new postons and veloctes fo all the patcles, the pogam epeats the cycle of updatng esultant foces and patcle acceleatons. C. Foces between Patcle and Lqud Phase The patcles used n the systems ae small (Stokesan) patcles. In ths case, the foces elated to the moton of a Stokesan patcle fallng unde gavty though a quescent lqud can be modeled by applyng the foces nvolvng: Buoyancy foces, F buoy F buoy = m g. (6) Whee m f s the mass of lqud dsplaced by the patcle, g s the acceleaton due to gavty. Dag foce, F dag f F dag = 6πµR p v. (7) whee μ s the lqud s vscosty, R p s the patcle adus, v descbes the elatve velocty of the patcle wth espect to the lqud. Hydodynamc esstance foce, F ess Consdeng the dynamc aspects of the thnnng of the ntevenng lqud flm between an a bubble and a patcle suface, the shot-ange hydodynamc esstance foce s pesent n the system. The expesson of ths foce s shown as Eq. (8), known as the Taylo equaton, whch gves a good ageement wth the expemental data wthn the ange of one bubble adus [30]. F ess 6πµ = H 2 R p v. (8) Whee s the vecto that jons the cente of the bubble wth the cente of the patcle, H s the dstance between patcle and bubble sufaces. The value of hydodynamc esstance foce tends to each nfnty as suface dstance appoaches zeo. In ode to avod ths numecal poblem, a cut-off dstance (H cut-off ) 0.5 nm has been establshed. Ths numbe s somehow abtay but takes nto account the lmt between classcal and quantum levels. Thus, ths foce was evaluated at zeo fo any dstances smalle than H cut-off. paametes and theefoe t povdes a lage vesatlty n the study of the attacton between hydophobc bodes [13, 17]. ( H / λ ) exp KR p = (9) whee K s elated to the maxmum stength of the hydophobc foce, whle λ s the length that ndcates how fast the hydophobc foce decays wth patclebubble dstance. The cut-off value of H cut-off s also used hee. The hydophobc foce appoaches ts maxmum value when suface dstance s less than 0.5 nm. A patcula focus n the pesent wok s placed on the effects of changng the values of K and λ on the captue effcency. E. Nomal Contact Foce The elastc foce F ne s modeled usng Hooke s law whch s popotonal to the nomal contact stffness k n and to the ovelap of the two sphecal sufaces δ n,.e. The ovelap s calculated as δ n F = k δ. (10) ne n n =. (11) d R 1 R 2 whee d s the dstance between two sphees (patcle-patcle o patcle-bubble) centes, and R 1,R 2 ae the espectve ad. Eq. (10) s only used n the case of ovelap less than zeo, othewse the elastc contact foce s equal to zeo. F. Compute Smulaton Paametes Compute smulatons wee pefomed utlzng the popetes of the mateals and flud shown n Table 2. The paametes of these mateals wee defned to be as close as possble to the physcal expements pefomed [31]. Fou dffeent values of decay length, λ, between 1 μm and 1 nm wee selected, snce ths ange coves any possble values epoted n the lteatue [4], and theefoe the effect of the hydophobc foce stength on the knetcs of captue of patcles by the bubble could be bette undestood. D. Foces between Patcle and Bubble Hydophobc foce, F h Thee s a lage vaety of models (see Table 1.) n the lteatue avalable fo hydophobc foces between bubbles and patcles. Howeve, thee s no unvesally accepted expesson. In the pesent eseach, the exponental fom was appled as t depends on two
TABLE II. COMPUTER SIMULATION PARAMETERS Patcles Numbe of patcles 200 Patcle densty 2.6 10-3 kg/m 3 Patcle adus 3.3 10-5 m Patcle stffness 100 N/m Bubble Numbe of bubbles 1 Bubble adus 1.0 10-3 m Bubble stffness 5.0 10 4 N/m Lqud Wate densty 1.0 10 3 kg/m 3 Wate vscosty 1.0 10-3 Pa s Hydophobc Foce Constant K 6.1 10 2, 6.1, 6.1 10-2, 6. 1 10-4 N/m Decay length, λ 10-6, 10-7, 10-8,10-9 m Othes Space dmenton 5 mm 5 mm 5 mm Tme step 1 ns Run tme 1s Fgue 2. The nomalsed numbe of patcle-bubble contacts as a functon of tme fo thee ndvdual systems consst of 200, 300 and 650 patcles. The decay length λ = 1 μm, constant K = 6.1 10 2 N/m. III. RESULTS AND DISCUSSIONS A. Influence of Patcle Concentatons and Intal Dstbutons on Patcle-Bubble Contacts In ths secton we pesent the nvestgatons consdeng the effect of the patcle concentatons and the andom dstbutons aound the cental bubble on the numbe of collsons aganst the bubble as a functon of tme. The esults of thee dffeent smulaton uns coespondng to 200, 300 and 650 patcles n the systems ae shown n Fg. 2. Each smulaton was caed out unde the same ntal condtons and usng the same values of the decay length and constant K n Eq. (9) (10-6 m and 6.1 10 2 N/m espectvely). It s obseved fom Fg. 2 that the smulatons fo the thee dffeent patcle concentatons poduce almost dentcal esults. Ths suggests that the 200 patcle smulaton s an appopate appoxmaton patculaly fo 650 patcle system n whch the patcle concentaton s vey close to expemental system. It s woth notng that due to the lmtaton of the sze of DEM systems the lage numbe of patcles nvolved, the longe the computatonal tme equed to complete a specfc gven value of eal tme. Theefoe, a system made of 200 patcles was found to be sutable fo the task of estmatng the knetcs of collson of patcles aganst the bubble. Undestandng of the nfluence of the ntal andom dstbuton of patcle postons on the collson esults s desable. Fg. 3 shows the compason of the effects of thee dffeent ntal patcle postons on the numbe of patclebubble contacts. Although the andomly geneated patcle postons befoe the smulaton ae dffeent, the vaatons Fgue 3. The nomalsed numbe of patcle-bubble contacts as a functon of tme fo thee ndvdual systems of dffeent ntal patcle postons. The decay length λ=1 μm, K=6.1 10 2 N/m. between the confguatons ae only notceable at the begnnng of the smulatons. Howeve, the fnal numbe of patclebubble contacts s nsenstve to the ntal condton. B. Effect of the Hydophobc Foce The nfluence of the hydophobc foce stength (usng Eq. (9)) on the attachment of patcles was nvestgated by changng the values of constant K and decay length λ. Fou dffeent values of K, whch detemnes the maxmum stength of the hydophobc foce, angng between 6.1 10 2 and 6.1 10-4 N/m, wee used n the smulatons. These values coespond to the maxmum value of the hydophobc foces equal to 2.0 10-2 and 2.0 10-8 N espectvely. Fou dffeent values of the decay length λ equal to 1.0 10-6 m, 1.0 10-7 m, 1.0 10-8 m and 1.0 10-9 m wee consdeed.
a b c d Fgue 4. Effect of hydophobc foce stength on the nomalsed numbe of patcle-bubble contacts as a functon of tme. Fgue 5. Effect of values of K on the nomalsed numbe of patcle-bubble contacts as a functon of decay length λ. Fgue 6. Effect of buoyancy foce on the nomalsed numbe of patcle-bubble contacts as a functon of tme.
Fg. 4(a-d) pesents the nomalsed numbe of patclebubble contacts as a functon of tme. Each cuve was obtaned fom sepaated smulaton caed out unde the same physcal condtons. Thee dffeent egmes can be obseved fo all cases plotted n Fg. 4(a-d). Ths s n ageement wth the esults epoted by Moeno-Atanaso [24] as expected. The numbe of patcle-bubble contacts dung the fst egme nceases qute slowly wth the tme. Ths s a consequence of the ntal statonay state of the patcles and the gavtatonal foce beng a domnant foce n ths stage. Theefoe, the patcles eque a cetan ctcal tme to each the bubble suface. The second egme s chaactezed by a apd ncease n the numbe of patcle-bubble collsons. The numbe of contacts then stays constant, whch epesents the thd stage of the patcle-bubble captue evoluton. Due to the nsuffcent hydophobc foce to ovecome the gavtatonal foce some patcles do not collde aganst the bubble, as hghlghted by Moeno-Atanaso (2013). When compang the fou plots n Fg. 4, the effect of the magntude of the constant K and decay length λ on the captue of patcles by the bubble was found to be damatc, especally fo the case whch λ s equal to 1.0 10-6 m. The numbe of contacts deceases wth deceasng the decay length λ and paamete K. Moeove, the hghe the value of the paamete λ o K, the faste the tanston nto the second and fnal egmes. It can be concluded fom the smulatons that the stength of the hydophobc foce s a sgnfcant facto nfluence the patcle captue effcency. Fg. 5 exhbts the nomalzed numbe of patcle-bubble contacts as a functon of decay length. The data epesent the ultmate values of the numbe of contacts fo the cases of K equal to 6.1 10 2, 6.1, 6.1 10-2 and 6.1 10-4 N/m.. As shown n Fg. 5, although deceasng λ poduces a decease n the numbe of patcle-bubble contacts fo each case, t exets a much smalle nfluence on the captue effcency. Fo each decease n stength of K (o the ato of maxmum value of hydophobc foce to the gavty) by two ode of magntude, the captue effcency deceased by 42.6% fo λ=1 10-6, 15.6% fo λ=1 10-7, 8.5% fo λ=1 10-8 and 12.5% fo λ=1 10-9. Inteestngly, each lne seems to show an exponental elatonshp between captue effcency and decay length λ. Howeve, ths elatonshp needs to be futhe nvestgated. C. Effect of the Buoyancy Foce The fnal step n ou study was to nvestgate the ole of the buoyancy foce on the captue of patcles. A compason between cases of the pesence and absence of buoyancy foce fo two dffeent values of the stength of the hydophobc nteactons s pesented n Fg. 6. The esults show that the tend of the cuves does not sgnfcantly change and the numbe of contacts eaches almost the same value n both scenaos of λ = 10-6 m, K = 6.1 10 2 N/m and λ = 10-6 m, K = 6.1 N/m. Ths s due to the fact that the magntude of the buoyancy foce s vey small (the value of the buoyancy foce nomalzed to the patcle weght s equal to 0.38). In contast, the atos of maxmum hydophobc foces to the patcle weght n these two cases ae 5.2 10 6 and 5.2 10 4. Theefoe, t s expected that the hydophobc attacton would have a moe pepondeant ole on the captue of patcles than the buoyancy foce. Howeve, the buoyancy foce esulted n a clea delay n the evoluton of the numbe of contacts. Ths s due to the fact that the buoyancy foce whch acts on the same lne wth gavty but n the opposte decton may slow down the patcle movng downwads to collde aganst the bubble. Ths phenomenon obvously affects moe sgnfcantly on those patcles whch ae located aound the uppe hemsphee of the bubble. When the patcles slde aound the bubble to the bottom hemsphee, most of them wll fall due to gavty ndependently of any buoyancy foce. CONCLUSIONS Thee-dmensonal DEM compute smulatons of patclebubble nteactons n a quescent medum have been epoted. Any conclusons dawn fom the smulaton data have been estcted to a monodspese patcle system. The qualtatve behavou of the 200 patcle system obseved hee has been shown to be qualtatvely coect fo hghe patcle concentatons whch ae close to the cases epoted n the lteatue. The hydophobc foce between the patcle and the bubble was estmated though a sngle exponental decay law whch depends on a constant K elated to the stength of the hydophobc foce and a decay length λ. It has been shown that nceasng the value of K o decay length λ can poduce hghe patcle captue effcency. Despte ths, t s obseved that the evoluton of the numbe of patcle-bubble contacts slows down by ntoducng the buoyancy foce of patcles, and gets faste by nceasng the magntude of the hydophobc foce paametes (K and λ). The esults pesented n ths pape consttutes the fst numecal analyss of the nfluence of the stength of the hydophobc nteacton usng the descpton of the sngle exponental law on the collson of patcles aganst bubble, and t s expected to be helpful fo the futue detemnaton of geneally vald expesson of hydophobc foce. REFERENCES [1] H. J. Schulze, Physco-Chemcal Elementay Pocesses n Flotaton. Elseve, Amstedam, 1983. [2] J. A. Fnch and G. S. Dobby, Column Flotaton. Pegamon Pess, Oxfod, 1990. [3] J. Ralston, D. Fonaseo, and D. R. Hayes, Bubble patcle attachment and detachment n flotaton, Inte. J. Mne. Pocess., vol. 56, pp. 133-164, Apl 1999. [4] A. V. Nguyen and H. J. Schulze, Collodal Scence of Flotaton. Macel Dekke, New Yok, 2004. [5] A. V. Nguyen, J. Nalaskowsk, and J. D. Mlle, A study of bubblepatcle nteacton usng atomc foce mcoscopy, Mne. Eng., vol. 16, pp. 1173-1181, Novembe 2003. [6] B. V. Dejagun and L. Landau, "Theoy of the stablty of stongly chaged lyophobc sols and of the adheson of stongly chaged patcles n solutons of electolytes", Acta Phys. -chm., vol. 14, pp. 633-662, 1941. [7] E.J.W. Vewey and J.T.G. Ovebeek, Theoy of the Stablty of Lyophobc Collods. Elseve, Amstedam, 1948.
[8] R. H. Yoon and S.A. Ravshanka, "Applcaton of extended DLVO theoy: III. Effect of octanol on the long-ange hydophobc foces between deodecylamne-coated mca sufaces," J. Collod Inteface Sc., vol. 166, pp. 215-224, August 1994. [9] R. H. Yoon and L. Mao, Applcaton of extended DLVO theoy, IV. Devaton of flotaton ate equaton fom fst pncples, J. Collod Inteface Sc., vol. 181, pp. 613-626, August 1996. [10] M. D. Ralston, D.W. Steeples, K. T. Spkes, and J. Bla, Nea-suface thee-component sesmc data acquston usng gdly nteconnected geophones, Intenat. Mtg., Soc. Expl. Geophys., 71st Annual Meetng, San Antono, TX, pp. 1411 1414, Septembe 2001. [11] H.-J. Butt and M. Kappl, Suface and Intefacal Foces. Wley-VCH Velag GmbH & Co. KGaA, Wenhem, 2010. [12] J. N. Isaelachvl, Intemolecula & Suface Foces. Academc Pess, London, 1985. [13] J. N. Isaelachvl and R. Pashley, The long-ange hydophobc nteacton decayng exponentally wth dstance, Natue, vol. 300, pp. 341-342, Novembe 1982. [14] R. H. Yoon, D. H. Flnn, and Y. I. Rabnovch, Hydophobc nteactons between dssmla sufaces, J. Collod Inteface Sc., vol. 185, pp. 363-370, Januay 1997. [15] W. A. Ducke, Z. Xu, and J. N. Isaelachvl, Measuements of hydophobc and DLVO foces n bubble suface nteactons n aqueous soluton, Langmu, vol. 10, pp. 3279-3289, May 1994. [16] N. Ishda and K. Hgashtan, Inteacton foces between chemcally modfed hydophobc sufaces evaluated by AFM The ole of nanoscopc bubbles n the nteactons, Mn. Eng., vol. 19, pp. 719-725, May-July 2006. [17] P.M. Claesson, C.E. Blom, P.C. Hede, and B.W. Nnham, Inteactons between wate stable hydophobc Langmu Blodgett monolayes on mca, J. Collod Inteface Sc., vol. 114, pp. 234-242, Novembe 1986. [18] P. M. Claesson and H. K. Chstenson, Vey long-ange attactve foces between unchaged hydocabon and fluoocabon sufaces n wate, J. Phys. Chem., vol. 92, pp. 1650-1655, Mach 1988. [19] N. Ishda, T. Inoue, M. Myahaa, and K. Hgashtan, Nano bubbles on a hydophobc suface n wate obseved by tappng-mode Atomc Foce Mcoscopy, Langmu, vol. 16, pp. 6377-6380, May 2000. [20] R. R. Dagastne and L. R. Whte, Foces between a gd pobe patcle and a lqud nteface: II. The geneal case, J. Collod Inteface Sc., vol.247, pp. 310-320, Mach 2002. [21] R. R. Dagastne, D. C. Peve, and L. R. Whte, Foces between a gd pobe patcle and a lqud nteface: III. Extacton of the plana halfspace nteacton enegy E(D), J. Collod Inteface Sc., vol. 269, pp. 84-96, Januay 2004. [22] A. H. Englet, S. Ren, J. H. Maslyah, and Z. Xu, Inteacton foces between a defomable a bubble and a sphecal patcle of tuneable hydophobcty and suface chage n aqueous soluton, J. Collod Inteface Sc., vol. 379, pp. 121-129, August 2012. [23] R. Maxwell, S. Ata, E. J. Wanless, and R. Moeno-Atanaso, Compute smulatons of patcle-bubble nteactons and patcle sldng usng Dscete Element Method, J. Collod Inteface Sc., vol. 381, pp. 1-10, Septembe 2012. [24] R. Moeno-Atanaso, Influence of the hydophobc foce model on the captue of patcles by bubbles: A computatonal study usng Dscete Element Method, Adv. Powde Tech., vol. 24, pp. 786-795, July 2013. [25] Y. Tsuj, T. Kawaguch, and T. Tanaka, Dscete patcle smulaton of two-dmensonal fludzed bed, Powde Technol., vol. 77, pp. 79-87, Octobe 1993. [26] P. A Cundall and O. D. L.Stack. A dscete numecal model fo ganula assembles. Geotechnque, vol. 29, pp.47-65, Mach 1979. [27] P. Attad and S. J. Mklavcc, Effectve spng descpton of a bubble o a doplet nteactng wth a patcle, J. Collod Inteface Sc., vol. 247, pp.255-257, Mach 2002. [28] S. Goldman, Genealzatons of the Young Laplace equaton fo the pessue of a mechancally stable gas bubble n a soft elastc mateal, J. Chem. Phys., vol. 131, p. 184502, Novembe 2009. [29] F. Y. Fage and P. A. Langston, Integaton schemes and dampng algothms n dstnct element models, Adv. Powde Tech., vol. 15, pp. 227-245, 2004. [30] T. H. Fan and O. I. Vnogadova, Hydodynamc esstance of closeappoached slp sufaces wth a nanoaspety o an entapped nanobubble, Phys. Rev. E, vol. 72, p. 066306 (1-9), Decembe 2005. [31] G. Bounval and S. Ata, Packng of patcles on the suface of bubbles, Mne. Eng., vol. 23, pp. 111-116, Januay 2010.