From the Henry KnIIrb School of MiIIe.t,Cohlrlbkl Uni.ersity. New York (U.S.A

Estimation of the effective number of - CH2 - groups in long-chain surfaceactive agents.)

By I. J. Lin, B. M. Moudgil, and P. Somasundaran

With 4 figures and 3 tabks(R«eived May 15. 1973)

methylene groups in the chain and upon thesubsequent evaluation of the effective chain-length of any similar surfactant with the helpof the relationships. Surface active propertiesthat are considered here are micellization. hemi-micelle formation at solidjliquid interface. sur-f:\ce tension lowering and adsorption at solution/gas interface.

Introduction

The chainlength or surface active agents is animportant parameter governing processes suchas flotation, flocculation, roam separation andactivated comminution and related interfacialphenomena such as adsorption and h<:nli-micellization. The solution properties such assurrace tension and micellization are also govern-ed largely by the length of the chain or thesurractants. In the case or most commercialsurrace active agents, however, the effectivechain length is different rrom the actual length.The difference arises rrom the fact that com-mercial surfactants often contain one or moreside chains and aromatic groups and in somecases even perfluoro groups in the chain in-creasing their hydrophobic character, or un-saturated bonds and substituted polar groupsincreasing their hydrophilic character. It isimportant to have a knowledge or the effectivechainlength or the surface active agents and thecharacteristics that determine it since it is theeffective chain length that is or more practicalrelevance than the actual chain length. In thispaper, major structural modifications that affectthe effective chain length are considered andseveral methods ror determining it are presented.

A. MicellizationMicelles of surfactants are formed in bulk

aqueous solution above a given concentrationfor each surfactant and this concentration knownas the critical micelle concentration (CMC) isrelated to the number of -CH2-groups in astraight chain homologous series by the relation

(1.2).CMC = A*exp(~--~~). [1]

Where A is a constant, W ~ is the electrostaticfree energy of micellization per molecule, n isthe total number of methyl end group andmethylene chain groups, 4>. is the van der Waalscohesive energy of.nteraction per -CH2-group,k is the Boltzmanns constant and T is theabsolute temperature. Eq. [1] for simple n-alkylamphiphiles can be rewritten in the form

logCMC = a - bn. [2]

Where a is

W~

Metlaods

The methods are based upon the identificationof the relationships between various surfaceactive properties of homologous series of n lkylsurfactants and the actual number of methyl and

*} Presented at the IOlst annual AIME meeting inSan Francisco. February 1972.

log A + 2.303 kT

cJ.and b is 2.303 iT'

...

An interesting application or this method isin the estimation or the effect or the position ofthe polar group in the hydrocarbon chain.Values for CMC's of sodium alkylsulrates con-taining a total of 8 to 18 CH2 and CH3 groupsbut with the sulrate groups attached at variouspositions are given in table 1. The CMCdatafor alkyl sulfates with sulfate group attached tothe terminal-C'H2-group (position I), can nowbe used to evaluate the effective chain lengthsof other sulrates. Towards this purpose the CMCvalue of alkyl (1) sulfates is first plotted as afunction of chain length as in fig. I. The chain-length corresponding to the CMC or othersulfates is then read as the effective chainlength.For example. as illustrated in fig. I, using a CMCvalue of 4.9 x 10-4, an effective length of 16.4is obtained for heptadecyl (2) sulfate. The effec-tiv~ chain length of other sulfates evaluated inthis manner is presented in table I. It can be

If W ~ is independent of the chainlength, therelationship between log CMC and n will belinear and it is in fact reported to be so (3).Lin and Metzer (4) determined the values of aand b for the homologous series of straight-chain alkylamine hydrochlorides using the dataof Ralston and Hoerr (5). The values obtainedfor a and bare 1.252 and -0.265 respectively..Eq. [2] for alkyl amine hydrochlorides is there-fore

log CMC = 1.252 - 0.265 n. [3]

Effective chainlength of any other surfaCtantsystem could now be obtained by determiningits CMC and substituting it in eq. [3]. Forexample, from the data of Lin and Metzer forCMC's of ethane-saturated and propane-satur-ated dodecylammonium chloride solutions val-ues of 12.35 and 12.80 were respectively obtainedfor the eff~tive number of -CH1-groups. Theequivalent increase in the number of -CH2-groups due to the dissolution of ethane is there-fore 0.35 units and that due to the dissolution ofpropane is 0.8.

-,.w-tOo.

IIS4..".

~..6.0...

10" -,~'4"-Otl4.-,~'4S1-0-

-,~'45S-0-

~'447-0-

-,~'444-0-

-,~'4?2-0POO

-,~'4"-On1.

1otta'4 s?-O 251.

\..\\

,o-~

.,

~\.I6

to-.J

'b~~c:. .....-t::-10-.' I I , 1+-+ I I

8 10 12 11# 16 18 20Total number OF- CHz- Groups, n

Fig. t. Plot of aitical mi~lIc oon=ttration as a func-tion of the total number of -CHz-groups in varioussodium alkyl sulfates

TOTAL MJMBER OF -CHz- GROUPS INTHE ALKYL CHAIN,n

Fig. 2. Plot of critical mi~lle concentration of alkyl-trimethylammoniwn bromides as a function of the totalnwnber of -CH l-grouPS in the alkyl dlain

4f1J

micelles form to the number n. of -CHI-groupsin the chain has been developed earlier (11).

l?g HMC = a' - bin [4]

where

and

t/I.b' ..2.303 kT

A' is a constant, KT~ is the electrostatic adsorp-tion energy, and ().. is the van der Waals cohe-sive energy for the transfer of a -CH2-groupfrom bulk water into a hemi-micelle. If hemi-micelle concentrations of long-chain salts in ahomologous series are known, the vaJues of a'and b' for that series can be calculated. Effective.chainlength of any other surfactant can then beeasily evaluated if hemi-micelle concentrationof it adsorbed on the same solid under the samecondition of pH, ionic strength etc. is known.Dick, Fuerstenau and Healy (12) recently usedthis method to determine the effective chainlength of various alkyl benzene sulfonates. Firstthey obtained a plot for herni-micelle concentra-tion of alkyl sulfonates as a function of thechain length using the data of Wakamatsu andFuerstenau (13) for the adsorption of sodiumdodecylsulfonate, sodium tetradecylsulfonate,and sodium hexadecylsulfonate on alumina atpH 7.2 and a concentration of 2 x 10-3 moleflKNO3. The values obtained for hemi-micelleconcentration of five dodecylbenzene sulfonateisomers were then placed along the plot ob-tained for straight-chain alkyl sulfonates andthe effective chainlengths of the isomers wereread from the plot. They observed the effectivechainlength to vary from a value of 15.2 to13.5 as the position of the benzene ring contain-ing the sulfonate group was shifted from oneend of the dodecyl chain to the center of it. Thisvariation clearly showed the effect of branchingof a surfactant chain on its interfacial activity.The results also showed that the addition of ?benzene ring to the chain is equivalent to theaddition of J.2CH2 groups to the chain. Thisis in excellent agreement with the effectivenessof the benzene ring estimated by other techni-ques and thus proves the validity of the methodof obtaining effective chain length using hem:-

seen that the shifting of the sulfate group awayfrom the terminal -CH2-group does affect theeffective chain length significantly. For example,despite the fact that sodium diheptylsulfonatecontains a total of 14 -CH2-groups its surfaceactivity is apparently equal to only that of astraight chain sulfate containing about eleven-CH2-groups.

It can also be seen that the relation.~hipbetween CMC and the number of -CH2-groups in the chain is linear for alkyl sulfateswith the ~SO4Na group in positions otherthan 1. This makes it possible to evaluate thechainlength of any alkyl sulfate with any memberof the homologous series as a basis.

Another interesting application of this techni-que is in the evaluation of the effect of coilingof long chained surfactant in water. Surfactantsmolecules which contain more than ab..)ut16 -CH2-groups are stated to coil aroundthemselves (1). While in aqueous solution suchcoiling should produce an effective decrease inthe chainlength. CMC data for alkyl trime-thylammonium bromides (8) are plotted in fig. 2as a function of chainlength. Non-linearity ofthe relationship between CMC and chainlengthfor longer chains is possibly due to the coiling.Using th,e straightline portion of the relationshipas a basis, the effective chain length of octadecyl-trimethylammonium bromide is found to be 11.2.Thus the coiling has resulted in the removal ofabout 0.8 CH2 group on the average from theaqueous solution. It must be noted that anyincrease in the tendency of the long-chainedsurfactants to form dimers, trimers etc. 'will alsoresult in an effective decrease in the chainlengthfrom the point of view of its micellization behav-ior. It might be noted that even though the aboveconclusions are on the basis of the micellizationbehavior of the surfactants, in general it isapplicable for other surface active properti~aiso.

B. H emimicellization

This method is based upon the observationthat surfactant ions adsorbed at the solid-liquidinterface associate to form two-dimensionalaggregates called hem i-micelles at a given ~ulkconcentration of the surfactant (9, 10). Thisconcentration, like the critical micelle concentra-tion. is determined largely by the chainlength ofthe surfactant. The following expression reiatingthe concentration, HMC, at which the hemi-

410 Colloid and Polymer Science, VoL 252 . No. .1

micelle concentration values. The method hasbeen further utilized successfully for determin-ing the effective length for the butane- andmethane-saturated dodecylsulfonate solution/alumina/air system (14).

three different values of surface tension aregiven in table 2. Using this result, values of0.46 and 0.72 were obtained for the effectivenumber of -CH2-groups in ethane and propanerespectively. Dissolution of ethane and propanein dodecylammonium chloride sulution is there-fore of the same consequence as lengtheningof the dodecyl ch~in by 0.46 and 0.72 CH2-group respectively.'

C. Surface tension lowering

Reduction in the surface tension of waterowing to the addition of organic compoundsbelonging to a homologous series is known to~how certain regularities as expressed. for ex-ample, by Traubes rule. Interpretation of thisphenomenon in terms of work gained on transferof the surfactant species from the bulk to thesurface yields a convenient way to evaluate theeffective chainlength of surfactants. If W.. andW..-1 are the work involved in the transfer of1 mole of a straight chain surfactant containingnand n - 1 -CH2-groups respectively

W.. - W..-1 = RTInS~

[5]

D. Adsorption at the liquid/gas interface

The previous method based on the Gibbsadsorption equation is simple and convenientbut inadequate for describing the adsorptionof long chain ions rigourously since it does nottake into consideration the electrical energyinvolved in the formation of an electrical doublelayer at the interface when the long-chain ionsare adsorbed there. Davies and Rideal (16) havederived a more rigorous isotherm for the caseof adorption of long-chain ionic surfactants.According to this the surface excess for the caseof a symmetrical electrolyte is given by

r = (B1/B2)Cexp[(W~ - Ze 1/1 o)/kT]

(Bl ) (W..~ Ze 1/1 0) [ ]1 + "Ii;..40 Cexp kT' 8

C.C:-I

= RTln [r.T.-IC.-1T.C.r.-1

where CS and C are the surface and bulk con-centration of the sudactants, r is the adsorptiondensity, and T is the thickness of the sudaceregion. W. - W.-1 has been found experimen-tally to be about 1.08 RT (7,15). Assuming T tobe constant. eq. [5] can be rewritten as

- r.c.-1 = 1.08[6]in -

c.r.-1The corresponding equation for the case of

a surfactant with an effective chainlength mwill be

I r. c. r"ln- = 1.08lm-n), i

f91

fl01J. = ~ . h - 1 ~-.!!.--(~ )1/2

.,.0 e Sin C.1/2 DRTI

Bland Bz are constants characteristic of theadsorption and desorption rates respectively,W d is the desorption energy of a hydrocarbonchain per molecule, Z is the valency of the polarhead, e is the electronic charge, D is the dielectricconstant of the medium, A is the area permolecule at the interface, Ao is the area permolecule for the case of a complete monolayer,1/10 is the surface potential, -n is the effectivenumber of-CHz-groups in the chain and 0';is the surface charge density. For normally

c r ' " L.J. ..The effective chainlength of a surfactant Can

be determined with the help of eq. [7] if thesurface excesses r .. and r. at bulk concentra-tions C.. and C. are known. Surface excessquantities can of course be calculated fromsurface tension data using the Gibbs adsorptionequation.

Lin and Metzer (4) used this method to calcu-late the effective number of -CH2-groups inethane and propane dissolved in dodecylam-monium chloride solutions. Results obtainedfrom surface tension data for dodecylammonium

411-"--'-

method has been further utilized successfullyfor determining the effective chain length for theethane- and prcOpane- saturated dodecylam-monium chloride solution/quartz/paraffmic gassystem (18).

encountered small values of C arid r. eq. [8]can be rewritten as

~ = ~exp(Ze"'o - W~

) + ~. [11]T 8. kT T-

Substitution of eq. [9] into the logarithmicform of [11] yields

Iog( ~-~ ) -~ '

r Talax

521 n

(B2 2.3kT = log ~

1200 n--2.3RT 2.3kTA1/2

Since A= 1/6.02 x 107 r

[12]

IOg(~ - r -, - ~ =lOg(t) - 23RT

1200( ')1/2] 1/2 [ 3]2.3 kT 6.02 x 10 n r .1

1/10 can be evaluated using eq. [10]. The lefthand side of eq. [13] can then be calculatedfrom experimental data and plotted as a fqnc-tion of rl/2to obtain n and log B1/B2 from theslope and the intercept respectively. If thesurface active molecule is not linear or if itcontains hydrophilic groups or double andtriple bonds in the chain, n will be actually theeffective number of -CH2-groups in the sur-factant ion. This method has been used here fordetermining the effective number of -CH2-groups in a dodecylammonium chain usingdata for its adsorption density at the liquid/airinterface. The plot obtained for

(£ - - C ) - Ze 1/1 0r r-

c 521 n

2.3kT

as a function of r1/2 is given in fig. 3. A valueclose to 12 obtained for n is in agreement withthe actual number of -'-CH2-groups in thechain. Jorne and Rubin (17) have similarlydetermined the effective number of -'-CH2-groups in a lauryl sulfate chain. They obtaineda value of 11.6 for n. If the polar head of anadsorbed surfactant is assumed to be fullyimmersed in the bulk liquid, all the carbonatoms in the chain cannot be considered to becompletely out of the aqueous environ~nlA value slightly smaller than the actual numberof carbon atoms in the chain can hence beexpected for n if it is evaluated using data for theadsorption at the solution/air interface. The

Zel/loFig. 3. Plot of log

(c/r - c/r max)- 23 kT

as a function of rl/Z for dodecylammonium dlloride andsodium lauryl sulfate

Discussion

Four methods based on various interfacialand colloidal properties of surfactants havebeen described for determining the effectivenumber of -CH2-groups in them. Ideally allthe four methods should yield the same number.

\

412 ~ 0IMl Pol)9ller Science, Vol. 252 . No.5

However, depending on the difference in effi-,ciencies with which a surfactant utilizes itsvarious hydrophobic parts in different inter-facial phenomena different values could beobtained for the effective number of -CH2-groups.

The value obtained for the effective numberis of course dependent upon the number ofhydrophilic groups and the number o( lipo-philic groups present in the surfactant moleculeor ion and the relative strengths of the groups.Surfactants have been classified in emulsiontechnology according to the balance betweenthe above two opposite kinds of groups referredto as hydrophile-lipophile balance (HLB). Thisclassification is on the basis of the total numberof various groups present in the chain and donot take into account the effect of their struc-tural position in it Thus, a straight chain sur-faCtant and a branched chain isomer of it wouldhave the same HLB value even though thebranched chain isomer would be less surfaceactive than its straight-chain counterpart. Hencethere is a need for a classification that is basedupon the surface activity of a compound A newclassification that is based on the effectivenumber -CH2-groups is proposed here. It isuseful for comparing the surface activity of twodifferent surfactants as wen as for deCmingchanges in surface activities brought about bythe presence of certain groups. For this classi-fication we deCme a hydrophobicity index (HI),which is equal to the ratio of the effectivenumber or -CHz-groups in a chain to theactual number in it A value greater than 1 dueto the addition of a group thus means an increasein the hydrophobic character of the surfactantdue to this addition and a value lesser than1 means a decrease in it. For example, substitu-tion of a single bond in the molecule with adouble bond or a triple bond win yield a HIvalues less than 1 since the latter are less non-polar than the former. Addition or a benzeneradical win however yield a value greater than1 because of its relatively high hydrophobicnature. Addition of -CF 2-grOUPS will alsoyield a value greater than 1 since perfluoroalkyl surfactants are much more surface activethan the corresponding alkyl surfactants (19).Shinoda, Hato and Hayashi (20) have recentlyshown that the critical micelle concentrationof a fluorinated surfaCtant is the same as thator a hydrocarbon surfactant whose chain length

lOitc

..,.1BI

Q.:)

~1610-'",

01111.:,..0

~1Z.0e~101~~ . B

//: I II II I ~s: B 10 12 14 16 18 ZD

Torol number OF.CHz- Groups, n

~ ,.,-~ --.'"-~- no. I 1.0

D 2 O.4. 3 O.u0 4 0110 I 0..... . 0.61. 1 084

. . 082Fig. 4. Plot ofeffectivc number of --CHz-groups in dievarious ajkylsu1fate chains as a function of the actualtotal number of -cHz-groups in dI~

is about 1.5 times that of the fluorocarbon chain.An HI value of about 1.5 is therefore obtainedfor a perfluoro surfactant. Simple lengtheningof a straight chain by the addition of -CHz-groups can normally be expected to yield anHI value of 1. When the number of carbonatoms in the chain is greater than about 16, thevalue will however be less than 1 due to thecoiling phenomenon discussed earlier. Additionof -CHz-groups as side-chains would alsoyield lower HI value since these -CHz-groups

are not as effective in contributing to the surfaceactivity as one added to elongate the straightchain portion of a surfactant (21). HI values ofvarious sodium alkylsulfates listed in table 1have been calculated by dividing the effectivenumber by the actual number and are given intable 3. It can be seen from the table that ahigher HI value is obtained by adding the-CHz-groups to the longer branch of thesur(actant rather than to the shorter one. A plotof the effective number as a function of the actUalnumber is given in fig. 4. The slope of the plotobtained for each homologous series is ameasure of the HI for that series. The difference

413Lin et ill., ~imation of the effective number of ~CH2.grm1ps in Iong-chabt surface active agents

Table 1. Effect of the position of polar head group in thehydrocarbon chain of sodium alkyl sulfates on theCMC (6) and the effective number of ~2-groups,n.rr. at 40 °C

Table 3. Hydrophobicity Index of various sodium alkylsulfates

fosition of Total number of Hydrophobicity-SO. Na Group -CH2.Qroups, n Index. HI

ncif

812141618

8101314151718

111415

141618

141519

1618

1314

1516

11111

0.9370.9500.9610.9640.9660.970.97

0.9360.9360.940

0.9140.9060.911

0.8850.8860.894

0.8750.877

0.8380.850

0.8260.825

812141618

8101314IS1718

111415

141618

141519

1618

1314

1516

0.1360.00860.0024O.(xx)580.(xx)16

0.1800.04950.00650.00330.00170.(XX)490.(XX)26

0.0290.00430.0022

0.005150.001720.(XX)45

0.006750.003400.(xx)33

0.00230.CXXJ7

0.01930.~7

0.00660.0042

812141618

7.59.5

12513.51.4.516.417.4

lC.313.114.1

12814.516.4

12413.317.0

14.015.8

10.911.9

12413.2

ttt ~

222222233)444SSS6617

1a

3

4

5

6

7

8

in values obtained for slope for various seriesclearly shows the effect of the position of the-SO.Na group on the hydrophobicity of thesurfactant ion. An examination of the plots inFigure 4 also shows that there is no measureable,~hange in slope as the chainlength is increased

in anyone given series. Thus there is no coilingexhibited even by octadecyl (1) sulfate chain.Considering the fact that the octadecyltrimethyl-ammonium bromide showed a coiling effect,this observation indicates an additional effectof the -SO4Na group.

A classification based on the effect of variousgroups and their position along with a know-

Table 2. Data for r ~-. C~-.lr ~-. C~-. and r ~-, C~-.lr ~-. C~-, at three different values of surface tension ofdodccylammonium chloride solution saturated with air (d-a), or ethane (d-e) or propane (d-p)

Surfaa.tension,dynesjcm

45SOS5

9.5 x 10-. 5.5 x 10-. 3.8 x 10-. 1.474.7xIO-tO 4.0xI0-tO 3.5xI0-to 6.2 x 10-. 33xI0-. 22xIO-. 1.61

2.0 x 10-. 13 x 10-. 13 x 10-. 1.70

1.862.132.29

1.59 2.00

ledge of the effective chainlength of variouscommonly-used surfactants should be helpful inmaking selection of surface active agents and inmodifying reagents and designing new ones forvarious interfacial processes such as flotation,flocculation, emulsification and solubilization.

Swrwnary

The effective length of long chain surfactants is amore im~rtant parameter than the actual chainlengthin governing various inteifacial and colloid phenomenasuch as adsorption and micellization and interfacial pro-~ Sucll as flotation, noccuJation, emulsification andactivated comminution. Various methods for determin-ing the effective length (or effective number of -cH2-groups) of ionic and non-ionicsurfactants are p~tedhere. These methods are based on treatment of experi-mental data for critical micelle concentrations, hemi-micelle concentrations, surface tension lowerings andadsorption densities at liquid/gas interface. A newclassification based on the ratio of the effective numberof --CH2-groups to the actual number, called thehydrophobic index, is pro~. The effects of variousstructural modifications and substitutions in the surfact-ant molecule or ion are discussed.

Z USanInIe'!f assung

Diec wirksame Uinge langkettiger Oberffiichenwirk-stofTe stellt einen wichtigeren Parameter dar als dieeigentliche Kettentange. soweit es die Beherrschung derverschiedenen Grenzflichen- und Kolloidalersch'einun-gen - wie Adsorption und Mizellisierung - und Grenz-fiichenvorginge - wie Flotation, Flockung. Emu1siftzie-rung und geladene Zerkleinerung - anbelangt.

In der vorliegenden Arbeit werden cine Reihe yonMethoden zur Bestimmung der wirksamen Lange (bzw.der wirksamen Anzahl -CH2-Gruppen) ionischer undnichtionischer OberflichenwirkstofTe dargestellt. Di~MC'thoden beruhen aufder Verarbeitung yon Versuchs-ergebnissen und ~aten rur kritische Mizellenkonzen-trationen, Halbmizellenkonzentrationen, Oberffiichen-spannungsverringerungen und Adsorptionsdichten ander Fliissigkeit/Gas-GrenzOache. Es wirdder VorschlageiDer neuen Klassiftzierung. aufgebaut auf dem Verhalt-nis zwischen der wirksamen Anzahl yon -cH2-Grup-pen und deren eigentlicher (nomineller) Anzabl. auchHydrophobieindex oder Wasserscheuigkeitszahl oderAbnetzungszahl genannt. gemacht. Die Auswirkungenverschiedener Abanderungen und Einsetzungen.im Mole-kill des OberflichenwirkstofTs oder dessen Ion werdenbesprochen.

References

1) S~a, K., Bulletin Chern. Sac. Japan 26, 101(1953); J. Pbys. Chern. ~ 1136 (1954).

2) PhiUips,J. N., Trans. Faraday. Soc. 51, 561 (1955).3) Osipow, L. 1., Surface Chemistry (New York 1962).4) Un, 1. J. and A. Metzer, J. Pbys. 0Iem. 75, 3CXX)

(1971).5) Ralston, A. W. and C. W. Hoerr, J. Amer. Chern.

Sac. 64, m (1942).6) EIJOlIS, H. C., J. Chern. Soc., 579 (1956).7) Mukerjee, P., Advan. Colloid Interfaoe Sci. I, 241

(1967).8) ShiniJda, K., T. Nakagawa, B. TlNIUI118AShi, and

T.Isemura, Colloidal Surfactants(New York, N. Y.1963j.9) SomasunJman, P. and D. W. Fuerstenau, J. Phys.

Chern. 70, 90 (1966).10) Fuerstenau, D. "'., T. W. Healy, and P. SMI/JS-

Undaran, Trans. AIME 219, 321 (1964).11) SlNnasundman, P., T. W. Healy, and D. W. Fuer-

stenau, J. Pbys. Chern. 68, 3562 (1964).12) Dick. S. G., D. W. Fuerstenau, aDd 7: W. Healy,

J. Con. Interf. Sci. 37,595 (1971).13) W~ T. and D. W. Fuerstenau, Advan.

Chern.. Sci. 79, 161 (1968).14) Smna.\1mdaran, P. and B. M. Moudgil, TheEfTect

of Dissolved Hydrocarbon Gases in Surfactant Solutionson Froth Flotation of Minera1s, 47th National ColloidSymposium, Ottawa, Ontario, June 18-23, 1973.

15) Lin,I. J.and P. SomasunJaran, J. Colloid Interf~Sci. 37, 731 (1971): Un 1. J., J. Phys. Chern. 76, 2019(1972).

16) Davies, J. Y. and E. K. RideDl, Interfacial Phenom-ena, 2nd Ed., p. 186 (New York 1963).

17) Jome, J. and E. Rubin, J. CoIL Interf. Sci. 38, 639(1972).

18) Lift, I. J. and A. Metzer, J. Pbys. Chao. (1973)to be pub1isbed.

19) SomasunJman,P.andR.D.Kulkarni, Trans.IMM82.164(1973).

20) ShiniJda, K., M. HalO, and T. Hayashi, J. Pbys.Chern. 76, 909 (1972).

21) Un, 1. J., J. P. Friend, and 1': Zimmels, ;. Con.Inter!. SQ. 45, 378 (1973).

Authors' addresses:

Dr. I.J. UnMineraI Engineering DepartmentT~bnion, Haifa (Israel)

Dr. B. M. MoudgilDepartment of Metallurgy and Material Scien~University of FloridaGainsville, Florida (USA)

P. SomaswrdaranColumbia Univ.New York, N. Y., 10027 (USA)