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Colloid Polymer Science Col lo id Polymer Sol 26 2:9 48 -95 5 (1984)

O LLO I D S IEN ETherm odynam ic consideration of the m ixed m icelle of surfactantsK. Motom ura , M. Yamanaka, and M. A ra tonoDepa r tment of Chemistry Faculty of Science Ky ushu Universi ty Fukuo ka Japan

A b s t r a c t : In confo rmi ty with the conclusion obtained previously the mixed miceUefor-mation of sur factants was treated thermody namically as the appearance of a macroscopicbulk phase with the aid o f the excess thermodyn amic quanti ties s imilar to those used forthe adsorbed f i lm. The comp osit ion of surfactant in the m ixed mice[ le and the therm ody -namic quanti ties o f miceUe formation were fou nd to be evaluated b y applying the the r -m od yn am ic equations derived. These equations were extended so as to be applicable toany kind of surfactant mixture. I t was sho wn th at the critical micelle concentration vs.com pos ition of surfactant curves form a diagram analogous to the phase diagram o f bi-na ry mixture. Ap plyin g the equation to the published data o n typical surfactant sys temsthis th ermo dyn amic ap proach w as pro ved to be useful to clarify the miscibil ity of surfac-tants in the micellar state.ey w o r d s :Critical micelle concentration mixed micelle surfactant mixture the rm od y-namics.

ntroductionT he m iscibility of suffactants in the m icellar state is

of cons iderable in te rest from the the rm ody nam ic andprac t ica l v iewpoin ts . I t was shown for the mixedmicelle m ade u p of l ike suffactant mo lecules that thedependence of the cr i t ical micel le concentrat ionCM C) on the co mp osit ion of suffactant is explicableby mean s of the idea l mixing theory [1-3] . R ecent ly atheo ry w hich treats m ixe d micel les as the regular solu-t ion was proved to descr ibe the behavior of mixedmice l le of unl ike suffac tan t m olecules [4-7] . H ow ev-er , there is considerable do ub t as to w he the r the regu-lar solut ion theo ry is applicable to th e m ixture o f sur-factant m olecules enforced to o rient at the surface withan extrem ely small radius o f curvature. Furth erm ore,i t is impossible to sup pose that , w he n an ion ic suffac-tant is considered, the mix ed micel le of suffactant ionssur rounded by an a tmosphere of counter ions in theaqueous m ediu m can be t rea ted as the regular so lu tion .Therefore , such a t rea tm ent is no t expec ted to be use-fu l . On the o ther hand, Motomura e t a l . [8-12]show ed tha t the mice l le behaves thermo dynam ica l lyl ike a macroscopic bulk phase whe n it s therm ody nam -ic quanti ties are given by th e excess the rm od yn am icquanti ties s imilar to those use d for the adsorb ed f i lm ofsuffac tan t. This v iew can imm edia te ly be e x tended to

the m ixed mice l le sys tems; therefore , one can der ivethe relat ions evaluat ing the m icel lar com pos it ion andthe therm ody nam ic quanti ties of m ice lle format ion .N ow the measu remen t o f the C M C as a func t ion o ftemp erature, pressure, and com pos it ion s expec ted toafford significant inform ation regarding the behav iorof m ixed micelles.In t h is paper , we p re sen t a t he rmody nam ic m e thodof obta in ing he c om posi t ion of sur fac tan t in the mix edmice l le and the thermodynamic quant i ty changesassociated with the m icel le forma tion. T he m iscibi li tyof surfactants in th e miceUar state is discussed by calcu-lat ing the composit ion of surfactant from publisheddata.Theoretical

Let us cons ider an aqueous so lu t ion comp r ised of n lmo les of the surfactant 1 and n2 moles o f the surfactant2. First , the surfactants 1 and 2 are sup posed n ot tohave a co m m on ion as the cons t i tuent ion ; the formerdissociates into V l ,a a-ions and Vl, c c-ions and th e lat terdissociates into v 2 , b b- ions and V 2 , d d- ions . The Gibbs-D uh em equat ion of the so lu t ion i s wi r t ten in the form- S d T V d p - n ~ d l ~ w - n a d l ~ - n c d l ~- n b d g b -- n d d g d = 0, 1)

8


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M o t o r n u r a e t a L , M i x e d r n i c e ll e o f s m f a c t a n t s 949w h e r e

/ / j= l ~ l ,j n l~ j = a , C ,a n d

n k = v 2 . k n 2 , k = b , d .

2)

3)W h e n a m i x e d m i c e l le o f th e s u r f a c t an t s is f o r m e d i nthe so lu t i on and i t s num ber o f m o les nm i s ve ry sm a l l ,t h e f o l l o w i n g e q u a ti o n m a y b e e x p e c t e d t o h o l d a t apos i t i on rem ote f rom the m ice l l e pa r t i c l e s :

4)- s d T + d p - Cwdl2w - c~dlx , - c r- cbdt - C d d l l d = 0 ,

w h e r e s i s t h e e n t r o p y p e r u n i t v o l u m e a n d ci i s then u m b e r o f m o l e s o f c o n s t i t u e n t i p e r u n i t v o l u m e .S in c e t h e m i c e ll e is s i m i la r in t h e r m o d y n a m i c b e h a v i o rt o t h e a d s o r b e d f i l m [ 8 - 1 2 ] , t h e b e h a v i o r o f t h e m i x e dm ice l l e i s p re sum ed to be desc r ibed in t e rm s o f t heexcess t he rm od yna m ic qu an t i ti e s de f ined in re fe renceto t he sphe r i ca l d iv id ing su r face wh ich m akes t heexcess num ber o f m o les o f wa te r ze ro , t h a t is ,n v - c . v v W = o , 5 )

w h e r e V w i s t he v o lum e ou t s ide t he d iv id ing su rfaces .N o w a n y m o l a r t h e rm o d y n a m i c q u a n t it y o f m i x e dm ice l l e is g iven by6)M = ( y _ y V W ) l n m ,

w h e r e Y is t h e t h e r m o d y n a m i c q u a n t i t y o f t h e s o lu -t i o n a n d y i s t h e o n e p e r u n i t v o l u m e a t a p o s i t i o nrem ote f ro m the m ice l l e pa r t ic l e s . S im i l a r ly t he num -ber o f j- i ons i n one m ixed m ice l l e pa r t i c le i s g iven by(7)M = n j - - c j V W ) / n m , j ----a b c d .

S ub t rac t i on o f equa t ion 4 ) m u l t i p l ied by V w f r o mequa t ion 1 ) and u se o f equa t ions 6 ) and 7 ) y i e ld t here l a t i on

8 )_ s M a T + v M d p - - N M d - N M d l 2o

n ' ~ c l , b - - N ~ d ~ a = O .Thi s i s t he fundam en ta l equa t ion desc r ib ing thet h e r m o d y n a m i c b e h a v i o r o f m i x e d m i ce ll es . I t s h o u l db e n o t e d t h a t t h e c o n d i t i o n o f e l e c tr o n e u t r a li ty h o l d sfo r t he m ice l l e pa r t i c l e s . Deno t ing t he m o la l i t i e s o f

s u r f ac t a n ts 1 a n d 2 w h i c h a r e d i s s o lv e d i n t h e f o r m o fions by rn l and m 2 , re spec t ive ly , and a ssum ing an idea lso lu t i on , t he t o t a l d i f fe ren t i a l o f t he chem ica l po t en t i a lo f t he i on i s wr i t t en a sd = - s i l T + v j d p + ( R T / m t ) d m ~

- ( R T / X ~ ) d X 2 , j = a , c 9)a n d

d l = - s k d T + v h d p + ( R T / m t ) d m ~+ ( R T / X 2 ) d X 2 , k = b , d , 10)

w here m t i s t he t o t a l m o la l i t y o f su rfac t an t smr----m1 + m 2 11)

and Xi i s t he m o le f rac t i on o f su r fac t an t i i n t he t o t a lsu r fac t an tX i = m i / m t , i = 1 , 2 . 1 2 )

F u r t h e r , t a k i n g i n t o a c c o u n t t h a t t h e m i ce U e f o r m a t i o ni s t rea t ed l i ke t he appea rance o f a m acroscop ic bu lkphase , ~ on the r i gh t s ides o f equa t ions 9 ) and 10 )c a n b e r e p l a c e d b y C M C i n a l i m i t e d c o n c e n t r a t i o nr a n g e n e a r t h e C M C . S u b s t i tu t io n o f e q u a t io n s 9 ) a n d10) i n to equa t ion 8 ) and rea r ran gem en t o f t he re su l t-i ng equa t ion l ead u s t o t he exp ress ion

[ v , , , , + V , , c ) X ~ + v 2 ,b + v 2 ,a ) X M 2 ] R T / C M C ) d C M C= A ~ v s d T + ~vdp R T [ v , , a + V l, c ) X ~ l X , )

- v2,b+ v 2 , d ) ( X M 2 1 X 2 ) ] d X 2 , 1 3 )w h e r e w e h a v e i n t r o d u c e d t h e m o l e f r a c t io n o f su r f ac -t an t i i n t he m ixed m ice l l e de f ined by

x? = N2I(N ( + N~)= N2IN~, i = 1 , 2 1 4 )a n d t h e t h e r m o d y n a m i c q u a n t i t y o f m i c e ll e f o r m a t i o nd e f i n e d b y

4 ~ y = y ~ I N P [ x 1 M P l , a Y a - [- " 1 )1 . Y c )+ X ~ ( " 2 . b Y b + " 2 . d Y d ) ]

= Y ~ I N P - [ X e y , X Y X 2 ] 9 1 5 )T h e r e f o r e , i f t h e C M C o f a n a q u e o u s s o l u ti o n o fsu r fac t an t m ix tu re i s m easu red a s a func t ion o f t he


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9 5 0 Colloid and Po lymer Science, VoL 26 2. No. 12 1984)

t empera ture , p ressure , and composi t ion of t he so lu-t ion , t he en t rop y of mice ll e format ion , t he vo lum e ofmice l l e format ion , a nd the c om posi t ion of sur fac tan tin the mix ed m ice ll e can be eva lua ted , respec tive ly , bythe relat ionsMa w s = - 1 2 X

+ v : X ~ ) R T / C M C ) O C M C I O T ) p , x : ,MA w v = v l X

+ v : X ~ ) R T I C M C ) O C M C l O P ) r , x ~ ,an d

an d= 1 2 2 X l v ,x + 1 2 x ) 24)

in plac e of the variables m~, 3;72, an d X ~. Th en equ ation13) i s rewri t t en in the form

1 6 ) R T / C l V l C ) d C M C = - [ 122X~+ 121xM2)/121122]A~vsdT+ [ v22~ + v , 2 ~ 1 1 2 , v 2 ] A ~ v d p

1 7 ) 2 5 )

X M = X 2 1 2 1 1 1 - - X ~ I C M C ) O C M C I O X 2 ) T , p ] I [ v 2 X ,+ 1 2 1 X 2- - (121 - - I " 2 )( X , X 2 / C M C ) ( a C M C / a X 2 ) r , p ] ,

1 8 )w h e r e

1"1 = 121, "{'- 121, 19)

122 = ~)2,b +" 1 )2 , d 9

an d

Fur ther , t he energy of m ice ll e forma t ion i s ca l cu la t edby use of t he re l a t ionMa w u = T A ~ s - p A ~ v .

N o w w e can obtain sufficient inform at ion to discussthe beh avior of mixed m ice ll e . I t is impor t an t t o no tetha t t he va lue of X ~ can be ca l cu la ted d irec t ly f rom theC M C v s. X 2 cu rv e w i t h o u t r e ly i n g o n t h e p re su m edmiscibi l i ty of surfactants in the m ixed m icel le . Ex am -in ing equat ion 18), how ever , X2M s fou nd to have a d i f-fe ren t va lue f rom X2 a t t he po in t wh ere O C M C / O X 2 =0 except for a sur fac t an t mix ture which meet s t hereq uir em en t 121 = vz. It m ay be desirable, as a rule , thata CMC vs . composi t ion curve of t he mixed mice l l eco inc ides wi th tha t o f the so lu t ion a t t he ex t rem um .H ere l e t us in t roduc e the var iab les def ined by

J ~ t ~ Plrnl ~ 2 H ' / ' 2= ( V l X 1 "o r-12:2X2)rnt,

5 : 2 = 1 2 x 2 / , , 1 X l + 1 2 x ) ,

w h e r e C l ~ C is re la te d to C M C b yC ] ~ / I C = ( 1 2 1X 1 -{ - 1 2 2 X 2 ) C M C . 26)

Thus , we h ave the re la t ionsA ~ s = - [121v2/ 122 f ~

+ 1 2 1 2 ~) ] R TIC 1 V IC ) OC l ~IC IOT)p ,2 ~ 2 7 )M

+ v,XM 2 ] R T / C M C ) O C M C I O P ) r , 2 2 , 2 8 )an d

20) Y 2 = 22 - X1X2/CIVlC) OC19IC/OX2)r ,p . 29)I t is seen f rom eq uat ion 29) tha t t he CM C vs . X~curve has an expec ted shape. F ur therm ore , t he ther -m od yn am ic quanti ti es of mice l l e format ion a re foun d21) to be est imated by plot t ing C1VIC against T and p.The se facts indicate that T , p , X 2 are th e variables suit-ab le for making d ear the s t ruc ture and proper t i es ofmixed mice l le .N ex t w e p ro ce ed t o co n s i d e r t h e ca se w h e re t h e su r -fac tan ts have a co m m on ion w he n they a re d i ssocia tedin the aqueo us solut ion. Sup posing that the su rfactant 1dissociates into vl , , a-ions a nd vl, c c-ions a nd the surfa c-tant 2 in to v2,u b-ions and v2,c c-ions, equ at ion 30) canbe der ived in a simi la r ma nne r to tha t o f equa t ion 13):

[ ( V I , a 4 - 1 } l ,c )X1M - i t - V 2 , b " [- 1 )2 , c ) X M ] R T I C M C ) d C M C= - a ~ s d T + AM w v dp + R T [ V ,, a + v ,, c ) X ~ / X 1 )

- v 2 ,b + v 2 ,D X l X 2 ) + 1 2 , cV 2 , c X - x 2 )/ X l X 2 (121,c X l " [ '- 1 2 2 ,c X 2 ) ] d X 2 , ( 3 0 )22) w h e r e C M C , X i , X ~ , an d A,~y are def ined by the ana-

23 ) logs of equ ations 11), 12), 14), an d 15), respe ctively.


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M o t o m u r a e t a l . , M i x e d m i c e ll e o f sw f a c t a n ts 951I t is c l e a r ly se e n t h a t e q u a t io n ( 3 0 ) i s t h e s a m e a s e q u a -t i o n ( 13 ) e x c e p t f o r t h e t h i r d t e r m i n t h e b r a c k e t o f th et h i r d t e r m o n t h e f i g h t s id e . T h e r e f o r e , b o t h t h e e q u a -t io n s c a n b e e x p r e s s e d i n t h e c o m b i n e d f o r m

121XM+ 122XM) R T / C M C ) d C M C= A M s d T + A M v d p + R T [ 1 2 1 ( x M / x 1 )

v ~ X ~ / X 9 + 6> , ,c 12 ~ ,~ x ~ - x 2 )/XlX2 121,cX1 J r 1 2 2 , d X 2 ) ] d X 2 ,

w h e r e 6 2 i s t h e K r o n e c k e r d e l ta d e f i n e d b y6 2 = 0 , d 4 = c

= 1 , d = c .T h e a b o v e e q u a t i o n d e s c r i b e s a n y m i c e l l a r b e h a v i o r o fa b i n a r y s u r f a c t a n t m i x t u r e . I t is s e e n f r o m e q u a t i o n( 31 ) t h a t , w h i l e M g i v e n b y t h ew s a n d A M v a r e s a m er e l a t i o n s a s e q u a t io n s ( 1 6 ) a n d ( 17 ), r e sp e c t iv e ly , X M sg i v e n b y t h e r e l a t i o n

A s e x p e c t e d , t h e v a l u e o f ~7M s s e e n to b e e q u a l to t h a to f X 2 a t t h e e x t r e m u m o f t h e C 1 V I C v s . X 2 c u r v e .I n o r d e r t o s h o w t h e u t il i ty o f t h is t h e r m o d y n a m i ca p p r o a c h , t h e c o m p o s i t i o n o f s u r f a c t a n t i n t h e m i x e dm i c e l l e i s c a l c u l a t e d f r o m p u b l i s h e d d a t a a n d c o n s i d -e r e d i n c o n n e c t i o n w i t h t h e m o l e c u l a r i n t e r ac t i o n b e -twe e n su r f a c t a n t s i n t h e m ic e l l a r s t a t e .

i s c u s s i on(3 1) T h e C M C o f s u r fa c t an t m i x t u r e w a s m e a s u r e d a s a

f u n c t i o n o f t h e c o m p o s i t i o n o f s u r f a c t a n t X 2 i n t h es o l u t i o n b y m a n y i n v e s t i g a t o r s . S e v e r a l b i n a r y m i x -t u r e s o f t y p i c a l s u r f a ct a n ts a r e t a k e n h e r e a s e x a m p l e s .F i r st , le t u s c o n s i d e r t h e s y s t e m o f p o t a s s i u m d o d e -( 32 ) c y l s u lf a te ( K D S ) a n d s o d i u m d o d e c y l s u l fa t e ( S D S ) a t3 1 3.15 K [ 13 ]; t h e f o r m e r su r f a c t a n t i s r e f e r r e d t o a s t h ec o m p o n e n t I a n d t h e l at te r as c o m p o n e n t 2 . T h e s u r-f a c t a n t s a r e 1 - 1 e l e c t r o ly t e s a n d h a v e t h e sa m e su r f a c -t a n t i o n ; t h a t i s ,

X M = X2[v l{1 - X i /C M C ) O C M C / O X 2 ) T ,p }- - r 1 2 , , c X l J r - 1 2 2 , d X 2 ) ] / [ 1 2 2 X 1 J r- 1 2 1 X 2- (121 v 2 ) X 1 X 2 /C M C ) O C M C / O X g ) r ,p- 6~Vl,cVi, / Vl,cX1 + v2,aX2)] 9 (33)

A g a i n X 2M s f o u n d n o t t o c o i n c i d e in v a l u e w i t h X 2 a tt h e p o i n t w h e r e OCMC OX2= 0 . O n t h e o t h e r h a n d ,u s e o f f i , ~ M , a n d C M C d e f i n e d b y e q u a t i o n s ( 23 ),( 2 4 ) , a n d ( 2 6 ) l e a d s u s t o t h e e x p r e s s io n( R T / C ~ v l C ) d C ~ v [ C = - - [ 1 2 2 2 1 M Jr 1 2 1 x M ) / v 1 1 2 2 ] / _ 1 M s d T

- {- [ l J 2 ~ [ M - { - 1 2 j ,M ) /v lV 2 ] A M v d+ RT[f~M11f 1--x M I 2 2

v 2 , a 1 2 1 ~ 2 ) ] d f ~ 2 . ( 3 4 )T h e r e f o r e , w h e n C l V l C is p l o t t e d a g a i ns t T , p , a n d X 2 ,t h e v a l u e s of AMs , AMv , a n d X 2M u r n s o u t t o b e e v a l u a t-e d n u m e r i c a l l y b y a p p l y i n g e q u a t i o n s ( 2 7 ) a n d ( 2 8 )a n d t h e r e l a t i o n

X 2M = X 2 - ( ) ( 1 X 2 / C 1 V I C ) ( 0 C 1 9 [ C / O X 2 ) T , p1 [ 1 - { S d V l , 1 2 2 ,d / ( 1 2 1 , c 1 2 2 X 1 - {- ] ) 2 , d 1 2 1 ~ '~ [ 2 )]

( 3 5 )

vl, a = Pl,c = V2,b= V2,c= 1. (36)T h u s , t h e v a l u e s o f X 2 a n d C i 9 I C a r e c a l c u l a t e d b y t h ef o l l o w i n g r el a t io n s d e r i v e d f r o m e q u a t i o n s ( 23 ) a n d(26);

X2 = X2 (37)a n d

C 1V IC = 2 C M C . ( 38 )N o w w e c a n o b t a i n t h e C 1 V I C v s . X 2 p l o t , w h i c h isi l lu s t r a te d b y t h e c u r v e 1 in f i g u re 1 . O n t h e o t h e r h a n d ,e q u a t i o n ( 35 ) r e d u c e s t o

f f M = f ( 2 - - 2 ( f ( l f ( 2 / C ~ / i C ) ( O C ] V l C / O f ( 2 ) T , p . (39)

1 7 . 0

6

16.0uu

1 5 . C I0 . 5 1 . 0

F ig. 1 . C r i ti c a l m ic e ll e c o n c e n t ra t io n v s . c o m p o s i t io n c u rv e s o f t h ep o t a s s i u m d o d e c y L su lfa teT.sod ium dod ecy l su lfa te sys tem a t313 .15 K [13] : 1 . C M C vs . X2 curve , 2 . C M C vs . X~ curve


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952 Colloid and Polymer Science, Vol. 26 2. N o. 12 1984)

Differentiating graphically th e C19IC vs. X2 curve a ndma king use of eq ua t ion (39) , the com posi t ion of sur -fac tan t 2 ~ in the m ixed mice l le i s evalua ted num er i -cally; the results are sh ow n in the form of C19IC vs. 2 ~curve, in dicated by 2, in figure 1. It is seen that th e valueof X2M s close to that o f 22 tho ugh th e form er is smallerthan the lat ter over the entire range of composit ion.Taking into account that both the surfactants haves imi la r CM C values and the i r com m on ion i s the sur -factant ion , the a bove fact suggests that KD S and SDSm ix ideal ly in the micel lar s tate. In addit ion, f igure 1m ay be regarded as an ana log of the phase d iagram,because i t expresses the relat ionship betw een the com -posi t ion of surfactant in th e m icel le and that in the solu-t ion at equil ibrium.It is now interest ing to see wh at effect the coun ter-ion of surfactant exerts on the CM C vs. com posit io ncurves. Taking up the copper dodecyl sulfate(Cu(DS)2)-sodium d o ef t sulfate system [13] wh ere

Vl ,a = l )2 ,b = 22 , c = 1 and v~,~= 2 , (40)the fol low ing elat ions are derived in a similar man ner:

2 X 2 / 3X 1 + 2X2 ) ,C]~/iC (3X1 + 2X 2)C M C,

41)(42)

andX 2 = X 2 - [ 2 1 2 2 ( 4 2 , + 322

/ ( 2 2 + , ) c [ c ] a c M c / a 2 2 ) r . p . (43)

15 0

v

EE 1 0 . 0

5.0

i i i i i i i

0 0 .5 1 .0

Fi g . 2. C r i t i c a l mi c e l l e c onc e n t r a uo n v s . c ompos i t i on c u r ve s o f t hec oppe r dod e c y l s u l f a t e - s od i um d ode c y l s u l f a te s y s t e m a t 303 .15 K[13] : 1 . CIVIC vs . ) (2 curve , 2 . CIVIC vs . X ~ curv e

These equat ions a re fo und to be som ew hat d if fe ren tfrom equations (37), (38), and (39). Figure 2 show s theCM C vs . X2 and C M C vs . X~ curves obta ined f romthe experim ental data at 303.15 K. I t is seen fromfigures 1 and 2 th at the difference betw een 2~ and 2 2values of the Cu(DS)2-SDS system is significantlylarge com pared wi th tha t of the KDS-SDS sys tem.Therefore , we can conclude tha t the enhan ced concen-t ra tion of Cu 2+ ion in the m ixed mice l le makes theCMC value decrease remarkably.As an al ternat ive to the ab ove system s, there is a sys-tem com posed o f two surfac tan ts of which the surfac-tant ions are different while the counterions are thesame. C ho osin g sodium tetradecy l sulfate (STS) as thepartner of so dium dod ecyl sulfate [14], equations (37),(38), an d (39) can l ikewise be us ed to evaluate th evalues of 22, C19IC, and 2 ~ . In f i ~ re 3, the ClqIC vs.22M curve is com pared wi th the C M C vs . 22 curve a t320 .15 K. The shape formed by these two curvesseems to be an appreciably th ick cigar type w he n com -pared w i th tha t of the KD S-SDS sys tem. Accordingly ,we m ay say tha t the mixing of STS and SDS in themicel lar state is less ideal than that of KD S and SDS,thoug h they d i ffer in the nu mb er of carbon a toms onlyby two .Inspecting figures 1, 2, and 3 of the system s com -posed o f two surfac tan ts car ry ing the co m m on on , i t i sseen that the surfactant of the smaller CM C value hasthe e nhanc ed comp osi t ion in the micel le . Fur ther , thediagrams depicted in the f igures are foun d to be of acigar type and useful to exam ine the difference in themiscibi l i ty of surfactants am on g the m ixed m icel les.

20.0

v... 15.02

EU


6/8

Mo tomura e t a l. , M ixed mice l l e o f sur fac tant s 953

T h i s k i n d o f b e h a v i o r m a y b e e x p e c t e d fo r th e b i n a r ym ix tu re o f s im i l a r non ion ic su rfact an ts .H e r e l e t u s c o n s i d e r th e h e x a o x y e t h y l e n e d o d e c y le the r C12 EO)6) -dodecaoxyethy lene dod ecy l e the rC1z EO)12) system. S ince the system is speci f ied byP l a ~ - ~ P 2 b ~ - ~ l a n d 1 21, =1 ~ 2 , d ~ 0 , 44)

equa t ions 23 ) , 26 ) , and 35 ) a re w r i t t en a sX 2 = X 2 , 45)C 1V IC C M C , 4 6 )

a n d~M ~ 2 - - (21X2/C~v~C)((~C~/iC/a22)T,p 9 4 7 )

I t s h o u l d b e n o t i c e d t h a t e q u a ti o n s 4 6 ) a n d 4 7 ) a r es l i gh t ly d i f fe ren t f rom the co r re spon d ing equa t ions o ft h e K D S - S D S s y s t e m . F u r t h e r m o r e , i t i s i m p o r t a n t tono te t ha t J~2 , C191C, and X ~ a re i den t i ca l w i th X2C M C , a n d X 2M, re spec t ive ly . In f i gu re 4 a re sho w n theC1VIC vs . X2 and C1VIC vs . X ~ cu rves o b t a ined f romexper im en ta l da t a a t 298 .15 K [2 ] . As expec t ed , t hecu rves a re seen to fo rm a c iga r shape . Th i s re su l t m ays u gg e st, w h e n c o m p a r e d w i t h t h o s e o f t h e K D S - S D Ssys t em , t ha t t he m ice l le o f t hese non ion ic su r fac t an t sbehaves l i ke an i dea l m ix tu re . I t i s now necessa ry t oc o n s i d e r t h e s y s t e m c o m p o s e d o f s ur f a ct a n ts w h i c h d on o t h a v e a n y c o m m o n i o n.

E m p l o y i n g c o p p e r d o d e c y l s u l f a t e a n d s o d i u mte t radecy l su l fa t e a s t he com ponen t s , t he fo l l owingre l a t ions a re de r ived in a s im i l a r m ann er a s above :f 2 = 2 X 2 / 3 X 1 + 2 X 2 ) ,C ] V [ C = ( 3 X 1 + 2 X 2 ) C M C ,

4 8 )49)

a n d2 1 2 2 / C M C ) O C g iC /O 2 2 ) ,,p . 5 0 )

I t i s w or thw hi l e t o recogn ize t ha t equa t ion 50 ) i s fa i r l yd i f fe ren t f rom equa t ion 43 ) fo r the Cu DS)2 -SD Ss y s t e m . B y a p p l y i n g th e r e l at io n s t o t h e e x p e r i m e n t a lda t a a t 320 .15 K [14 ] , we ob ta in t he pecu l i a r d i ag ramw h i c h i s d r a w n i n f i g u r e 5 . T h e s y s t e m s h o w s t h edev ia t i on f rom idea l i t y i n such a w ay tha t t he C lVIC vs .J~2 and CIVIC vs . J?~ cu rves have the m in im um po in ta t w h ich theTg co inc ide . I t i s c l ea r ly seen tha t X ~ i ss m a l l er t h a n X 2 a t a h i g h e r c o m p o s i t i o n t h a n t h a t o f t h em i n i m u m p o i n t a n d l a r g e r a t a l o w e r c o m p o s k i o n .T h e r e f o r e , t h i s n o n i d e a l b e h a v i o r o f th e m i x e d m i c e ll ecan be sa id t o be a t t r i bu t ab l e t o t he fo rm a t ion o f t hec o m p o u n d C u T S )2 o f w h i c h t h e C M C v a lu e is a p p re -c i ab ly sm a l l . We no t i ce t ha t t he d i ag ram in f i gu re 5resem bles t he phase d i ag ram o f t he azeo t rop i c sy s t em .S ince t he a zeo t ropy i s c lo se ly re l a t ed t o t he i n t e rac t i onb e t w e e n c o n s t i tu e n t m o l e c u l e s i n t h e s y s t e m [ 1 5 ], t h ea b o v e b e h a v i o r m a y b e a c c o u n t e d f o r a p p a r e n t l y b ythe s t ronge r m o lecu la r i n t e rac t i on be tween Cu DS)2

0 200 . 1 5

-6EEu(.. ) 0.10

0. 05 , , , , i , , , ,0 0 .5 1 .0

Fi g . 4. C r i t i c a l mi c e l l e c onc e n t r a t i on v s . c om pos i t i on c u r ve s o f t heh e x a o x y e t h y l e n e d o d e c y l e t h e r - d o d e c a o x y e t h y l e n e d o d ec ~ d e t h e rsys te m a t 298.15 K [2] : 1 . C M C vs . X2 curve , 2 . CIVIC vs . X~ curv e

5.0

4.06EE,,, 3.0J


7/8

and STS . A ccord ing ly , i t is pa r t i cu l a r ly i n t e re s t i ng tosee how the m ixed m ice l le o f su r fac t an ts cha rac t e r i zedby a specia l molecular in teract ion behaves.L e t u s c o n s i d e r t h e m i x t u r e o f t e t r a o x y e t h y l e n eoc ty l e the r C8 EO)4) and sod ium dod ecy l sul fa te a t298 .15 K [2 ]. H ere t he re l a t i ons app l icab l e to t he ex -pe r im en ta l da t a a re

= 2 x = / x , + 2 x 2 ,C 2 r = X , + 2 X 2 ) C M C ,

5 1 )

a n d5 2 )

15.0

x ~ = x 2 - . ~ 1 x 2 / c M c ) a c M c / a 2 2 ) r , p , 53)w hich are de r ived f rom equa t ions 23 ) , 26 ) , and 35 ) .Th us , we ob ta in t he p lo t s o f C1VIC aga ins t ~72 and X2s h o w n i n f i g u re 6 . A s e x p e c te d , t h e y p a s s t h r o u g h t h ec o m m o n m i n i m u m . T h i s a z e o t r o p i c b e h a v i o r c a nundoub ted ly be a sc r ibed to t he a t t rac t i ve i n t e rac t i onb e t w e e n t h e e t h e r o x y g e n o f C8 EO)4 a n d t h e a n i o n i cg r o u p o f S D S [ 2 , 1 6 , 1 7 ] . H o w e v e r , n o m i n i m u m i so b s e r v e d o n t h e C M C v s . c o m p o s i t io n c u r v e o f t h eoc t ao xye thy lene dode cy l e the r C12 EO)8) -sod iumdo dec yl su l fa te sy stem at 298.15 K i l lust ra ted in f igure7 [2 ] . C om par ing f igu re 7 w i th f i gu re 6 , t he C M C ra t i oo f C~2 EO)8 to SDS i s found to be ex t rem e ly sm a l lc o m p a r e d w i t h t h a t o f C8 EO)4 to SDS. I t i s wel lk n o w n t h a t t h e n e g a t iv e a z e o t r o p y t a k es p l a c e i f t h e

2

7 . . . 10 . 06EE

uu 5.0

954 Colloid an d Polymer Science, VoL 26 2. N o. 12 1984)

r i

0 0.5 1.0~ 2 ~Fig . 6 . Cr i t ical micel le conce nt rat ion vs . com posi t ion curves o f thet e t r ao x y e t h y l en e o c t y l e t h e r - so d iu m d o d ecy l su lf a te sy s t em a t298.15 K [2]: 1. CM C vs. , t~2 curve , 2. C1V[C vs. X~ curve

1 5 . 0

2

0 0.5 1.0

Fig. 7 . C r i t ical micel le concent rat ion vs . com posi t ion curves o f theo c t ao x y e th y l en e d o d ecy l e t h e r - so d iu m d o d ecy l su l fa te sy s t em a t298.15 K [2]: 1. ClVIC vs. J 2 curve, 2. ClVIC vs. 2 ~ curve

7._ 10.05EE

-to

5.0

m olecu la r i n t e rac t i on be tween com ponen t s i s a t t rac -t iv e a n d t h e p u r e c o m p o n e n t s h a v e s i m i l a r v a p o r p r e s -su re va lues [15 ] . The re fo re , we can conc lude tha t t heazeo t rop i c behav io r o f t he m ice l l a r sys t em i s de t e r -m ined by the m o lecu la r i n t e rac t i on be tween su r fac -t a n ts a n d t h e r a ti o b e t w e e n t h e i r C M C v a lu e s. C o n s e -quen t ly , t he pos i t i ve azeo t rop i c behav io r i s t o bee x p e c te d f o r a b i n a r y m i x t u r e f o r m e d f r o m s u r fa c t an t sbe tw een w h ich the m o lecu la r i n t e rac t i on i s repu l s ive .Ac tua l ly , such a behav io r was obse rved in cases o ff l u o r o c a rb o n s u r f a c t a n t - h y d r o c a r b o n s u r f ac t a n t s y s-t e m s , t h o u g h i t w a s c o m p l ic a t e d b y t h e f o r m a t i o n o ftwo types o f m ixed m ice ll es [5 , 7 ] .I t is ev iden t f rom th e above d i scuss ion tha t t he m i s -c ib i l i ty of surfactants in micel les can be e lucidatedth rou gh the app l i ca t i on o f t he equa t ions de r ived in t hep rev ious sec t i on to expe r im en ta l re su l t s . Fu r the ri n f o r m a t i o n r e g a r d i n g t h e s t r u c tu r e a n d p r o p e r t i e s o fm i x e d m i ce lle s w i l l b e p r o v i d e d b y m e a s u r i n g t h eC M C a s a f u n c t i o n o f t e m p e r a t u r e a n d p r e s s u re a n dt h e n e v a l u a t i n g t h e e n t r o p y , v o l u m e , a n d e n e r g y o fm ice l l e fo rm a t ion wi th t he a id o f t he equa t ions .Referen ces

1 . Sh inoda K 1954)J Phys Che m 58:5412 . Lan ge H, Beck KH 1973) Kol lo id Z u Z Polym ere 251:4243 . C l in t JH 1975)J C S Fara day 1 71 :13274 . Rubingh DN 1979) ed) Mit tal KL , So lu t ion Ch emis t ry o f Sur-factan ts , P lenum, N ew York 1 :3375 . Sh inoda K, Nom ura T 1980) J Phys Chem 84:365


8/8

Motomura e t aL Mixe d miceUe of sur factants 955

6. H ua XY, Ro sen M J (1982) J Co llo id In te r face Sc i 90:2127 . F u n a s a k i N , H a d a S (1 9 8 3) J P h y s C h e m 8 7 :3 4 28 . M o t o m u r a K , I w a n a g a S, Y a m a n a k a M , A r a t o n o M , M a t u u raR (1982) J C ollo id Interfa ce Sci 86:1519 . Y a m a n a k a M , A r a t o n o M , I y o t a H , M o t o m u r a K , M a t u u r a R(1982) Bull Che m Soc Jpn 55:27441 0. Y a m a n a k a M , I y o t a H , A r a t o n o M , M o t o m u r a K , M a t u u r a R(1983) J Collo id In te r face Sc i 94:4511 1. M o t o m u r a K , I w a n a g a S , U r y u S , M a t s u k i y o H , Y a m a n a k a M ,Matuura R (1984) Collo ids and Sur faces 9 :191 2 . Ya m a n a k a M , A r a to n o M , M o to m u r a K , M a tu u r a R ( 19 8 4)Collo id Pol ym er Sc i 262:3381 3 . M o r o i Y , M o to m u r a K , M a tu u r a R ( 1 97 4 )J Co l lo id I n te r f a ceSci 46:11114. M oroi Y, Nish ik ido N , M atuura R (1975) J Collo id In te r face Sc i50:344

1 5 . P r igo g in e I , De f a y R ( 19 5 4 ) Ch e m ic a l Th e r m o d y n a mic s , Ev e -r e t t D H T r an s, L o n g m a n s - G r e e n , L o n d o n 4 5 016. Jones M N (1967)J Collo id In te r face Sc i 23:3617. Sc hwug er MJ (1973)J Collo id In te r face Sci 43:491Rece ived May 4 , 1984;accepted June 14 , 1984

Authors ' address :K i ns i M o t o m u r aD e p a r t m e n t o f C h e m i s t r yFacul ty of Sc ienceKy u s h u Un iv e r s i ty 3 3F u k u o k a 8 1 2 , J a p a n

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