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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 Waalsforces. 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~ 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

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122

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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 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

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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

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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 .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]. Mackor [37]huestimated this interaction energy by considering the loss in entropy dvtt&(fue restriction of the molecules during the attachment. The expressionderlt-dcby Mackor is, Vs=8N_RT(1-Ho/~) (9) ~ where N - is the maximum number of sites availablefor adJqJ't1on;8 is the fractional surface coverage;and ~ is the adsorbedlayer thickM'.' The steric repulsion has therefore been calculated using the foBcwtrigexpression with ~ = 20 A. vs = raIL R7'(l- Ho/20)X Ac

It should be noted that Mackor's derivation is only valid atO.~dsorption densities when there is negligible lateral interaction betw--.

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126

EXPERIMENTAL

I Materials

(i) Alumina Linde grade alumina of 0.3 ~ size purchasedfrom Union Carbide Corpora- I tion had a BET surface area of 15 m2/g and an isoelectric point of 8.7 as deter- mined by electrophoresis.

~ (ii) Surfactant Sodium dodecylsulfonate (> 98% pure) purchasedfrom Aldrich Chemicals was used as received.

(iii) Inorganic reagents Sodium chloride (> 99% pure) (Aldrich Chemicals),hydrochloric acid and sodium hydroxide (Fisher Scientific Company) used were of reagentgrade.

Samplepreparation

The alumina was conditioned in sodium dodecylsulfonate solutions using a magnetic stirrer. After conditioning for 6 h. a small amount of the samplewas removed for the measurementof zeta potential. The remaining sampledwas centrifuged at 4000 rpm for 10 min and the residue was used for the bubble pick-up experiments and aggregationtests. The supernatant was centrifuged again at 5000 rpm for 10 min and divided into five portions and a portion was used to determine adsorption. a secondportion to measuresurface tension. a third portion for zeta potential samplepreparation and the remaining portions w.ereused for bubble pick-up and aggregationtests. '", Experiments

(i) Surface tension Surface tension measurementswere made using the Wilhelmy Plate method and a sandblastedplatinum sensor.Readings were taken at equilibrium.

(ii) Zeta potential The zeta potentials were measuredusing a Lazer Zee-Metermanufactured by Pen Kern, Inc.

(iii) Adsorption The residual concentration of sulfonate was detennined by a two-phase technique [38] and therefrom the adsorption was obtained.

(iv) Bubble pick-up (a) Equipment: A sketch of the equipment used is shown in Fig. 2. The sur- factant solution was held in a beaker (D). The alumina particles were con- "'1 ) :;§~ ,'i--!\

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127

BUBBLUI_CK UP APPARATU~ I

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(NOT TO SCALE)

A STAND B GLASSSPOUT C HAMILTONMICRO BURET SYRINGE '~ D GLASSBEAKER (50ml) E NARSHIGHE ML - B MICROMANIPULATOR

Fig. 2. Schematic diagram of bubble pick-up apparatus.

tained in a spout (B). Air bubbles were produced by meansof a micro buret syringe (C) which was attached to a micromanipulator (E) so that the syringe could be moved in vertical and horizontal directions as desired. (b) Procedure: An air bubble of given volume was conditioned in the sur- factant solution for 30 s, and then contacted with the equilitmated alumina for 30 s. The microsyringe with the bubble was then moved away from the spout and the bubble was released.The procedure was repeated100 times and the weight of the transferred particles was obtained after centzifUgingand dry- ing the contents of the beaker.

(u) Aggregation tests - About 0.2 g of the residue was redispersedin the surfactaat by stirring for r 15 min, after which the light transmissionat a given position in the suspension ~ was measuredas a function of time using a Brinkman probe.

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128

RESULTS AND DISCUSSION

I The bubble pick-up behavior of the alumina-dodecylsulfonate systemis il- lustrated in Fig. 3. Interestingly, tlle shapeof the curve is similar to that of tlle Hallimond cell flotation of quartz using dodecylamine shown in Fig. 1. A max- imum is obtained for the alumina "flotation" at.approximately 4 X 10-4 I kmol/m3 sulfonate.

I " I I I' 'I " NaDDS/ALUM1NA " T =2SoC .To1° C .010 pH = 6.2 .To0.1 I = 1 X10-2 kmol/m3

~ (I) (I) .008 I&JI&J .J.J ~,., (,)m -m ~:) C(m ~~ t:z~.!::~~:: B. 0 .006 0-ILO '\ ...>- \ \ xm ~ ~ B. .004 \ I&J:) / \ ~o \ I&J \ ~ (,) ~ .002 ~ , .,f', /°

~ 1 I I I IIII I I II I IIII I I c,.i"" ~"'t,~ / '~ 10-5 10-4 10-3 sf!' ~ ~ ,\ RESIOUAL SULFONATE, kmol/m3 ;' Fig. 3. Weight of particles picked up by 100 bubbles as a function of dod~lfonate ~~ concentration. Results for the sulfonate adsorption density, zeta potential aJMtaggregation 1~~ of alumina particles are given in Figs. 4, 5 and 6 respectively. ~~~ ,,'l To determine the interactions responsiblefor bubble--particle Blhesion, the ) bubble and the attached particle are assumedto be 5 A apart. ~ area of con- tact between the bubble and particle is estimated to be 10-14 .2 ~basedon geometrical considerations for a 0.3 J.Lmspherical particle with a 20 A thick adsorbedlayer interacting with a flat surface). The free energy changeresulting from tJ1etransfer of the dolie.:yl chain from the solid-liquid to tJ1eliquid-air interface, VH (calculated1lsingEq. (1) and the data for adsorption density and zeta potential) is given.. Fig. 7. This attractive interaction goesthrough a maximum as a result- a reduction in t/J~:l-L/G with increasein surfactant concentration. This reStion re- sults from the fact that the lateral associationof adsorbedsurf.-ant species

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~ 129

10-8 - '-- ~'- I , , "I I I I I , , " "i'1' NoDDS/ALUMINA ,I r.258C:t18C pH s 6.2:t 0.1 I . 1X10-21 0.lx10-2 kmol/m3 ,.;p.

10-9 ~ N E y "0 E f / / z~ r 0 ~la-1ol ~ , ' 0 cn 0 I.~~,~i <{ I&J t- <{ Z 0 ~ 10-11 ::> cn I 4./ 10-12L . I I I I IIII I I I I I IIII I I I 10-8 10-4 10-3 ~ RESIDUAL SULFONATE. kmol/m3 Fig. 4. Adsorption isotherm of dodecylaulfonate ODalumina at pH 6.28d ionic strength 1 x 10-2 kmol/m3 NaC. at the solid-liquid interface at high concentrations already ""S to the re- moval of a fraction of CH2groups from the aqueousenviro~nt and that the transfer of surfactant from the solid-liquid to the liquid-gas interface could result in the removal of only some of the remaining ~ groups. In contrast, at lower concentrations, in the absenceof lateral 8BOciation, a higher fraction of CH2 groups would be removed from the allleous environ- ment upon transfer of the surfactant coated surface to the ]XJIid-gas inter- face. Also shown in Fig. 7 is the energy changedue to steric r~ion, VB, cal- culated using Eq. (10). It can be seenthat Vs is not signific- compared to VH suggestingthat the system is not sterically stabilized. The double layer interaction, for reasonsdiscussed earlier,. is taken to be the averageof the valuescalculated using Eqs. (4) and (5). Ho~r, calculation of

~ 130 -60 *-T--r "Tetro.l,t NoDDSI ALUMINA T.25.C:t;,.C I -40 pH . 6.2 to.' I . ,.'O-z.kmol/m3 > - 2.0 e tf .J 4 0 I- Z I&J I- ~ 0 +20 A.

4 I- I&J +40 I /t:;o ~jc~ .. +60 ~ I +80 1:, . "'" III1 ,. I I IIIII I I I I IIIII I I I I IIIII, . I Ii'

0 10-6 10-5 10-4 10-3 ,10-2 .' RESIDUAL SULFONATE, kmol/m3 Fig. 5. Zeta potential of alumina particles at pH 6.2 and ionic strength 1 X 10-2 kmol/m: NaCl. NoDDS/ALUMINA 1 . 25.Ct 1.C pH . 6.2 t. O. 1 I . 1x 10-2 kmol/m3

~

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I&J I-u 0- 90 .,.oo.[:i U / Z

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10-5 10-4 10- 3 RESIDUAL SULFONATE. kmol/m3

~ .,

111

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+ 1 I I I I I I I . I I . I I I , , I I I I I I I I-LLLJ

10-5 10-4 10-3 10-1- RESIDUAL SULFONATE, kmol/m3 Fig. 7. Interaction energy due to transfer of adsorbed surfactant moleculea fnnn the solid- liquid interface to the liquid-air interface and interaction energy due to staii: repulsion. this interaction requires valuesfor the potential of both surfaceLThe poten- tial on the particle surfaceis assumedto be equal to the zeta potatial, which in turn was calculated from electrophoretic mobility data. The iNMentialon the bubble surfacecan be estimated from the surface tension data for the sys- tem (see Fig. 8) by fllSt calculating the adsorption densitiesat the liquid-air interface using the Gibbs equation [39], ~ rL/G = -(1/2o303RT)(d'Yid log CR)

and inserting th'",sein the Gouy-chapman equa~on, t18], rLl9 = -(1/F)(8 X 103 'NaekTC)1/2 sinh (ZFl/lbIRT] The bubble potential, 1/Ib,calculed in this manner is given in ~ 9. Values for ~ - 1/1 ~d VE-o calculated using Eqs. (4) and (5) respectively.e given in ~ig. 10. ..YE,the averagevalue for the double layer interaction ~rgy, and VEV (= VE + Vv where Vv = 1.93 X 10-18 J) are also shown. VEV thus rep- resentsthe repulsive energy barrier that must be overcome befJB the particle can attach to the bubble. The overall interaction energy VT, i.e., Vs + VE + Vv + VH .given in

Fig. 6. Light transmission through alumina suspension as a measureof a88gation of alumina in dodecylsulfonate solutions.

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100

HoODS/ALUMINA 90 T z 25.C .t. 1. C I pH a 6.2 .t.0.1 I = 1x10-2 kmol/m' E "- 80 z E

I z I-f~-- - - -~.I2J ~~c~~~ .. Q 10 U) Z IAJ I- ~ IAJ 60 "-.q (,) ~ ~ " II. ~ " ~ 50 U) ~ ~',.. "~~.'~ "'_C8- 40 r :~.~ , "'- .."'-;~IL'~~_~' 30..,;'1'1111111111",,1 Iltllllll '11'11,,1'1'1"'1' 0 10-5 10-5 10-4 10-3 lor2 RESIDUAL SULFONATE. kmol/m3 Fig. 8. Surface tension of dodecylsulfonate solutions (supernatant from a_rption teats).

~

~ ':-- ' .,A 't~,-- " -.

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~ ,. .<2 K 0 , ~ ~~ >-- ". Vy ~ « , .~, ~ I&J Z +1 \. I&J '\ "", r-r '..' vE Z \ ",".":

~-~ +4 VE

10-5 10-4 10- 10-2 Fig. 10. Interaction energy due to overlap of electrical double layers of bubble and particle based on constant potential and constant charge assumptions. Also averageiDteraction '~ energy, VE and net interaction due to van der Waal's forces and overlap of d-.ble layers (distance of separation between bubble and particle assumedto be 5 A). Fig. 11 as a function of sulfonate concentration. The bubble pick-up curve for this system is also shown in the sameFigure. The trend of the wbble pick-up clearly matchesthat of VT with the concentration correspondingto the max- imum bubble pick-up correlating with that of maximum total inteJoaction energy. Homoaggregationof the alumina particles and changesin the size of the bubble can also contribute towards determining the amount of mineral levitated by the bubble. Under certain conditions alumina particles were ob- servedto aggregateand such an aggregationcan result in enhancedmineral pick-up at low aggregationand possibly in reduced pick-up at very high ag- gregation. Also, chargereversal of the type observedhere at hi~ surfactant

Fig. 9. Surface potential at the bubble surface (liquid-air interface) calculated using surface tension results, and Gibb's and Gouy-Chapman equations.

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D ~ U)- 0 I&J P""- VT ...I I . ~ ~ .., ::) m 0 ~ >-- C-' 0.006 g ~ "" U) z kI "" ...I 2 z- BUBBLE ... 0 0.004 ~ ~ u PICK-UP f c I&. "~~-'- _c_". j '--". ~ ) " "" 0 ...... z 0.002 ~ ~ \. I&J ~ 0 ~ 0

, "'" "I .". .. ,,1 "" ""I - 10-8 10-4 10-1 10-2 "~~ RESIDUAL SULFONATE. - . ."'°1/.. ~go 11. Correlation of bubble pick-up with total interaction energy, VT - VB + Vs + Vv VEo '~ concentration conceivably could produce a decreasein aggregation,leading to reduced attachment of mineral to bubbles. However, in the pr~t caseno such decreasewas observedat high surfactant concentrations (~ Fig. 6). Evidently the sum of interactive forces between bubble and particle is the major factor determining the extent of mineral pick-up by the oobbles.

SUMMARY f!

Mineral pick-up by bubbles was found for the alumina-dodecylsulfonate system to exhibit a maximum similar to that obtained often in the past for flotation. This behavior is quantitatively accounted for in this study in terms of the interaction energiesinvolved in the attachment of a particle to a bubble. The overall interaction energy is the sum of contributions from 1) the attrac. tive hydrophobic forces resulting from the partial transfer of dodecylsulfonate chains adsorbedat the solid-liquid interface to the liquid-ak interface; 2) the net repulsive forces (throughout most of concentration rangestudied) resulting from the van der Waals interaction, the overlap of the electrical

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135 double layers of the particle and the bubble and the steric repulsion between I the surfactant layers adsorbedon the two surfaces.For tlle presentsystem these forces are calculated from data obtained using some test samplesfor sulfonate adsorption density. electrophoretic mobility of alumina particles and surface tension of sulfonate solutions. Homoaggregationof particles also was monitored by measuringthe supernatant clarity of mineral suspensionsin I sulfonate solutions. The overall interaction energy calculated in thWmanner was found to-correlate with tlle results obtained for bubble pick-up for tlle ~ sametest solution; concentration corresponding to tlle maximum wbble pick- up correlated well with tllat for maximum interaction energy betweenbubbles and particles.

ACKNOWLEDGMENTS

-,-~ --=-C::~- ~ The authors wiShto acknowledgethe Chemical and ProcessEngineering .'\, Division of the National ScienceFoundation (ENG-78-25213) for supporting ~l~~ ~ ~ this research,and Dr. K.P. Ananthapadmanabhanfor helpful discussions.

REFERENCES

1 P. Sornasundaranand L.T. Lee, Separation Science and Technology, 16 (1981) 1475. 2 P. Somasundaranand L. T. Lee, XIV International Mineral ProcessingCongress, Toronto, 1982. 3 K. L. Sutherland and I. W. Wark, Principles of Flotation, Australian Institate of Mining and (Inc.) Melbourne (1955) 257. 4 P. Somasundaranand B.M. Moudgil, J. Colloid Interface Sci., 47 (1974) 290. 5 B. V. Derjaquin and L. Landau, Acta Physicochirn. U.R.S.S., 14 (1941) 633. 6 E.J. W. Verwey and J.Th.G. Overbeek, Theory of Stability of Lyopho- Colloids, '~ Elsevier, Amsterdam, 1948. 7 B. V. Derjaquin and S.S. Dukhin, Trans. lost. Min. Metall., -70 (1961) 221. 8 K.L. Sutherland, J. Phys. Chem., 52 (1948) 394. 9 L.R. Flint and W.J. Howarth, Chern. Eng. Sci., 26 (1971) 1155. 10 D. Reay and G.A. Ratcliff, Can. J. Chern. Eng., 51 (1973) 178. 11 P. Somasundaran and D.W. Fuerstenau, J. Phys. Chern., 68 (1964) 3562- 12 T. Wakamatsu and D.W. Fuerstenau, Adv. Chern. Ser., 79 (1968) 161. 13 M.J. Rosen, Surfactants and Interfacial Phenomena, Wiley-lnterscience, New York, 1978. 14 A. meier, E.D. Goddard and R.D. Kulkarni, Flotation, M.C. Fuerste~ (Ed.), Vol. I (1976) 117. 15 Th.F. Tadros, Adv. Colloid Interface Sci., 14 (1980) 144. 16 J. Lyklerna. Adv. Colloid Interface Sci., 2 (1968) 65. 17 I.J. Un and P. Somasundaran, J. Colloid Interface Sci., 37 (1971) 18 G.A. Parks in Riley and Skirrow (Eds.), Chemical Oceanography, Acadelilic Press, New York, 1975. 19 J.F. Scarnerhorn, R.S. Schecter and W.H. Wade, J. Colloid Interface ti, 85 (1982) 494. 20 B. V. Derjaquin, Disc. Faraday Soc., 18 (1954) 85. 21 R. Hogg, T.W. Healy and D.W. Fuerstenau, Trans. Faraday Soc., 62 (1934) 155. 22 G.R. Weiseand T.W. Healy, Trans. Faraday Soc., 66 (1970) 490.

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