,,;- -'; ... - ~, -.; A 1- Colloids and Surfaces, 8 (1983) 121-136 121 Elsevier Science Publishers B. V., Amsterdam Printed in The Netherlands A STUDY OF THE INTERAcrIONS BETWEE~CL~S AND BUBBLES IN SURFACTANT SOLUTION~ P. SOMASUNDt\RAN, P. CHANDAR, and K. CHARI Henry Krumb School of Mines, Columbia University, New York. N. Y. 10027 (U.S.A., ~ (Received 20 July 1983; accepted 27 July 1983) ABSTRACT Attachment of particles to bubbles involves various interactions resulti..- from electrical -;;:~~~~:;,:;::.~ double layer forces between the particle and the bubble (with adsorbed surfactant on both). van der Waals forces. the energy change due to the transfer of the hydrocarbon chains ad- ~L~.~~~~~ sorbed on the particle to the gaseousphase. and steric repulsion between .-orbed sur- factant layers on the two interacting surfaces. The energiesof these interactions for the alumina-dodecylsulfonate system are calculated using data for the zeta potential of the mineral. the surface tension of the solutions and the surfactant adsorptioa density. Estimated total interaction energy is correlated with the results of bubble ~-up experi- ments. Like flotation. mineral pick-up by the bubble goes through a maxiD81Das a func- tion of surfactant concentration and this behavior is satisfactorily accouDt..J (or by the present treatment. INTRODUCriON Flotation is a complex processthat is the result of many chemiCaland '~ hydrodynamic phenomenain a system that contains solid partides, liquid, and gasbubbles in a state of varying turbulence. While certain indiYdlal phenomena such as adsorption are fairly well understood, the manner in wbr.h these various phenomenainteract to yield flotation of a mineral hasDOt been fully established.Also, although the effects of most of the system ~erties on flotation are known, the mechanismsby which they control flotation have not been fully developed. Flotation of minerals has been shown in tile past to ex- hibit a surfactant concentration dependencesometimes characterized by a maximum [1-4]. This is illustrated in Fig. 1 for the quartz~cylamine system [1]. While the increasein flotation evidently is due to. hydro- phobicity imparted to the mineral surface by surfactant adsorpCi!>n,the de- creaseat higher surfactant concentrations can be due to surfactat adsorption in this range with a reverseorientation as well as to mutual re~ion between the bubble and the particle, which will be similarly chargeduIDr tJ1esecondi- tions. Excessiveaggregation between particles, making them t8Ocoarse for levitation, or a decreasein bubble size in this concentration rmJle also can lead 0166-6622/83/$03.00 e 1983 Elsevier Science Publishers B. V. i; J l: 122 ./ I 0 &&J t- o« 0 ..J IA.. # .. ~ . ~t;;'- too,. -~ "- ~-~ :."~ ~, DCA - HCI CONCENTRATION. kmol/m3 FiC. 1. Flotation of quartz in a Hallimond Cell uling dodecylamine hydrocl8ride as collector [1]. to a reduction in flotation. In this stucjy, an attempt is made to ~titatively account for the above flotation behavior in terms of various inteiations be- ~ tween the bubble and the particle. In order to accomplish this, 1fIe,bubble- particle attachment processis treated as a heterocoagulation pr~. While some models have been developedfor coagulation between solid )8rticles, no model taking all the interactions into account exists for aggregatDt between a solid particle and a bubble. Aggregation between particles and ldbles under flotation conditions is complicated by the presenceof significant hydrophobic interactions between the bubble and the surfactant ions adsorbedon the particle. The bubble-particle at~hment processis treated here essentilily in terms of DL VO theory [5,6], modified to include both the hydropho18 attractive interaction arising from the transfer of surfactant chains from tile solid-liquid interface to the liquid-air interface and the repulsive interactioo due to pos.- sible steric hindrance between adsorbedsurfactant layers on the mteracting bubble and on the particle. As indicated above, hydrodynamic forces also significantly affect the over. all flotation process[7-10]. In order to isolate such effects, tbe"flotation" behavior was studied here using a bubble pick-up technique unlk quiescent / j ~ oj.. .Jr t conditions. In addition, tJ1ebub~le:::::particl~ !!-l~tioIIS~y zed at a :iii~il~~~~~Ce...Q!; 5 A-;-where hy~odynamic iDi::enctiona ~.fI'iJ.i ,I The system selectedfor the present study was alumina(O.3..-)-dodecyl sulfonate about which considerableinformation already ~1tIie literature [11-13]. Also, most importantly, in this work all the plO~ftJ1e system I required for tJ1eestimation of various intera:tion e~~ mctlt>eorrelation with flotation were measuredfor the systemusing the 8me ~les. THEORY The energy changesassociated wiUl Ule attachment of a pat*! to a bubble in Ule flotation processarise from Ule overlap of Ule electricalick'blelayers of the particle and the bubble. from van der Waalsforces, from ~fer of surfactant adsorbedon the particle to the gaseousp~, and~dition. from steric hindrance or volume restriction of the 8urfactant:~es on the interacting surfaces[7.14-16]. Expressionsdeveloped to estiate'the magni- tude of these interactions are discussedbelow. Hydrophobic interactiom The free energy, VH, involved in the transfer of dodecyl eiiairof the sur- factant adsorbed on the solid surface to the gaseousphase ~attachment of the bubble to the particle can be estimated as V = rS/L",S/L-L/G X 12A \ .,." , ..~ 01'"df:>:t° H "'CH. ~ co: 6.~'j ( ~'" ~."~\) II ) rS/L is adsorption density at the solid-liquid interface, ~G is the tzansferenergy per mole of CH2 groups from the solid-liq\i1~the liquid- gasinterface and Ac is the areaof contact between bubble aD~cle. A1112 " hydrocarbon groups including the end group are consideredctbelidentical in this treatment. .S/L-L/G can be estimated to be .L-L/G - .L-S/L (2) CHI CH. CHI where superscriptsL-L/G and L-8/L represent the transf~z groups from bulk liquid to liquid:-gas and solid-liquid interfaces ~ely. From past studies [17], .~L/G - -:-1.08RT. ,~/L can be estim88tusingthe 'J~ following adsOrptioniAotherm of dodecylsulfonate on alUm811{18 J : 1 _I v rS/L/(rS/L - raIL) ~ (C /55.5) exp - ( .!!. L-S/L + ~ -¥, .'~ max R [ RT CHI RT J (3) ~ ~~ where CR is the equilibrium concentration of sulfonate in kdliD3 and r is ~ the zeta potential of alumina, which is assumedto be the Sta~tential. rS/L is the maximum adsorption density and is estimated~4.5 X 10-6 m~m 2 from surface tension studies and packing consider.- [19]. Using ~ iiA 124, these expressions, V H can be calculated from data for adsorption density and zeta potential. ,I Electrical double layer interactions Expressionsfor the interaction energy of the overlap of double layers I [20-26] differ essentially in the choice of boundary conditions, l8nely, con- stant chargeor constant potential of the interacting surfacesduring their mutual approach. For the caseof constant potential at both surf~, the fol- """" lowing equation derived by Hogg et al. [21] is appropriate: ~-w = eala2(1/Ij + 1/1~) 21/111/12In '1 + exp (-KHo) \ E + 4(al + a2) (1/1I +I/I~) 1":: up (-.«Ho) In (1- exp (-2"Ho» .",..-," '"-:- (4) ~18::1,. -.-~~~~~- If, on Ute other hand, Ute chargedensities at boUt interfaces is a-.med to re- "- main constant, Ute equation derived by Weiseand Healy [22] usiugthe proce- .J~~~= dure suggestedby Frens and Overbeek [27] applies: VF,-o = ~-111 - Eala2(1/1~ + 1/1~) In (1- exp (-2"Ho» (5) 2(al +a2) ...~1" .\J~ ". Severalpublications have dealt with Ute choice of boundary conditions ,,''\. [27-30]. The constant potential condition prevails if electroch~ equi- ~ librium of the potential determining ions and adsorbing speciesbetween the ~ .(~~ bulk solution and the interface is maintained during the interac1D [30,31]. rr ~J' Alternatively the constant chargecondition would be more appr.."jate if the ~-~ ~- adsorption density of Ute speciesthat are responsiblefor the ch~ develop- .-~ ment at the interface is not regulated rapidly enough during the ~roach of ~ t' - the surfaces[30,32]. In a real system, both the chargeand the potential can change,particularly if the adsorbedsurfactant species"also contrilutes to the development of the surfacepotential. Under such conditions, the tEe of neither the constant potential nor the constant chargecondition will be valid, and an intermediate condition between these two extremes will .. more ap- propriate [33]. Accordingly, for Ute present treatment, the val~ C)fthe double layer interaction energy is taken as the arithmetic averageof the gues cal- culated usin:gequations 4 and 5. van der Wool'sinteractions Since the present system is comprised of three different mediafsolid, air and intervening solution), there will be an energy changeassoc~ with van der Waal'sforces when the bubble and particle are brought togelier [7]. For the caseof a fiat plate and a sphere(ap « ab), this free energydange is given by [34], Vy = -Aap/6Ho (6) ./ if,~ :~i1f:;.,." .. J: I where A is the Hamaker constant for the system,ap and ab ~1tl~ radii of the " particle and bubble respectively, and Ho is the distance of ~on. The Hamaker constant for the system is given by [35]. A = (~ - ~)(VA'PP - VAll) (7) , The values listed by Visser [35] of the Hamaker constants f«~na (App) and water (All) are 15.5 X 10-20 J and 4.38 X 10-20 J, r~tJ\I'ely. The value for air (Abb) is considered to be negligibly small. The Y8lItefor A is there- ~ fore -3.86 X 10-20 J. Interestingly. the Hamaker constantilffjfind to be nega- tive. suggestingthat in this casethe van der Waalsinteractim1!isr~wsive. The concept of a negativeHamaker constant and correspondingly;~ repulsive van der Waalsinteraction has been discussedby Visser [36).~8 Vv for the abovevalue of A is. -=~ .. "'o! '~~~":1_"..'1_'~' t ., Vv = 9.65 X 10-28/Ho (8) : - - ---=--~ -=-- Interaction due to steric repulsion When the interacting spheres(bubble and particle in thjs aNt bontain ad- sorbed layers, their adhesion will be sterically hindered, ~y due to the physical size of the molecules.Therefore the treatment of tb~egation phenomenon has to be modified to include a volume rest~tteim in the ex- pressionfor overall interaction energy [15,16].