Nonideal Mixed Monolayer Model - ACS Symposium Series (ACS

8 Nonideal Mixed Monolayer Model Paul M . Holland

Downloaded by STANFORD UNIV GREEN LIBR on February 15, 2013 | http://pubs.acs.org Publication Date: June 5, 1986 | doi: 10.1021/bk-1986-0311.ch008

Miami Valley Laboratories, The Procter & Gamble Company, Cincinnati, OH 45247

A generalized nonideal mixed monolayer model based on the pseudo-phase separation approach is presented. This extends the model developed earlier for mixed micelles (J. Phys. Chem. 1983 87, 1984) to the treatment of nonideal surfactant mixtures at interfaces. The approach explicity takes surface pressures and molecular areas into account and results in a nonideal analog of Butler's equation applicable to micellar solutions. Measured values of the surface tension of nonideal mixed micellar solutions are also reported and compared with those predicted by the model.

In mixed surfactant systems, physical properties such as the critical micelle concentration (cmc) and interfacial tensions are often substantially lower than would be expected based on the properties of the pure components. Such nonideal behavior is of both theoretical interest and industrial importance. For example, mixtures of different classes of surfactants often exhibit synergism (1-3) and this behavior can be utilized in practical applications (£5).In addition, commercial surfactant preparations usually contain mixtures of various species (e.g. different isomers and chain lengths) and often include surface active impurities which affect the critical micelle concentration and other properties. An important motivation for understanding the behavior of mixed micellar solutions is that equilibrium between the micelles and monomers establishes the chemical potential of the different surfactant species in bulk solution. In turn, these chemical potentials provide the driving force for processes such as interfacial tension lowering,contact angle changes and partitioning, and at equilibrium exert a controlling influence on physical properties at the solution interfaces of interest. Viewed from this perspective, the mixed micellar problem represents the crucial first step toward developing 0097-6156/ 86/ 0311 -0102$06.00/ 0 © 1986 American Chemical Society

In Phenomena in Mixed Surfactant Systems; Scamehorn, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8.

HOLLAND

Nonideal Mixed Monolayer Model

103

a g e n e r a l i z e d model f o r t h e b e h a v i o r o f n o n i d e a l mixed s u r f a c t a n t systems. Pseudo-phase s e p a r a t i o n models f o r t r e a t i n g mixed m i c e l l i z a t i o n have become i n c r e a s i n g l y u s e f u l s i n c e t h e e a r l y 1950's when they were f i r s t d e v e l o p e d f o r b i n a r y systems based on t h e assumption o f i d e a l m i x i n g i n t h e m i c e l l e s (7 Q). F u r t h e r developments f o r the i d e a l models have i n c l u d e d e x p l i c i t t r e a t m e n t o f monomer concent r a t i o n s and m i c e l l a r c o m p o s i t i o n s (9.) mixed i o n i c systems w i t h d i f f e r e n t c o u n t e r i o n s (10-11) two phase systems (12) and m u l t i component s u r f a c t a n t systems (12-14)· The e a r l i e s t a t t e m p t s t o e x p l i c i t l y t r e a t n o n i d e a l m i x i n g i n m i c e l l e s (13-16) were somewhat cumbersome and have n o t been w i d e l y used. However, a much more t r a c t a b l e a p p r o a c h i s based on t h e r e g u l a r s o l u t i o n a p p r o x i m a t i o n and was f i r s t a p p l i e d by Rubingh (17) t o a wide range o f n o n i d e a l m i x t u r e s . A l t h o u g h t h i s thermodynamic model was s p e c i f i c a l l y d e v e l o p e d f o r n o n i o n i c s u r f a c t a n t m i x t u r e s , i t was a l s o f o u n d to be u s e f u l i n d e s c r i b i n g t h e b e h a v i o r o f m i x t u r e s c o n t a i n i n g i o n i c s u r f a c t a n t s . Many subsequent t r e a t m e n t s o f n o n i d e a l s u r f a c t a n t m i x t u r e s have employed t h e r e g u l a r s o l u t i o n a p p r o a c h i n one form o r a n o t h e r . These i n c l u d e Ingram's e x t e n s i o n o f b i n a r y n o n i o n i c a p p r o a c h t o model s u r f a c e t e n s i o n s o f b i n a r y s u r f a c t a n t m i x t u r e s ( 1 8 - 1 9 ) Rosen and coworkers' d e t a i l e d t r e a t m e n t o f s y n e r g i s m i n b i n a r y systems ( 2 - 3 ) , a model i n c l u d i n g t r e a t m e n t o f n o n i d e a l b i n a r y i o n i c s u r f a c t a n t m i x t u r e s by Kamrath and F r a n s e s ( 2 0 ) , t r e a t m e n t o f b i n a r y a n i o n i c - n o n i o n i c m i x t u r e s by Scamehorn e t . a l . (21_) and a g e n e r a l i z e d model f o r n o n i d e a l multicomponent m i c e l l e s by H o l l a n d and Rubingh (14.)· Although the u n d e r l y i n g assumptions o f the r e g u l a r s o l u t i o n a p p r o x i m a t i o n a r e known t o f a i l i n a number o f t h e s e c a s e s (22-24) and i t has been c r i t i c i z e d on fundamental grounds (23 »26), i t does p r o v i d e the most t r a c t a b l e and u s e f u l way t o t r e a t n o n i d e a l mixed m i c e l l a r s o l u t i o n s i n many s i t u a t i o n s . The purpose o f t h i s p a p e r w i l l be t o d e v e l o p a g e n e r a l i z e d t r e a t m e n t e x t e n d i n g t h e e a r l i e r mixed m i c e l l e model (14) t o noni d e a l mixed s u r f a c t a n t monolayers i n m i c e l l a r systems. I n t h i s work, a thermodynamic model f o r n o n i o n i c s u r f a c t a n t m i x t u r e s i s d e v e l o p e d which c a n a l s o be a p p l i e d e m p i r i c a l l y t o m i x t u r e s c o n t a i n ing i o n i c surfactants. The form o f t h e model i s d e s i g n e d t o a l l o w f o r f u t u r e g e n e r a l i z a t i o n t o m u l t i p l e components, o t h e r i n t e r f a c e s and t h e t r e a t m e n t o f c o n t a c t a n g l e s . The use o f t h e pseudo-phase s e p a r a t i o n a p p r o a c h and r e g u l a r s o l u t i o n a p p r o x i m a t i o n a r e d i c t a t e d by t h e r e q u i r e m e n t t h a t t h e model be s u f f i c i e n t l y t r a c t a b l e t o be applied i n r e a l i s t i c situations of interest. 9


9

9

Theory The pseudo-phase s e p a r a t i o n a p p r o a c h has been s u c c e s s f u l l y a p p l i e d i n d e v e l o p i n g a g e n e r a l i z e d n o n i d e a l multicomponent mixed m i c e l l e model (see 14.) a n d i t i s i n t e r e s t i n g t o c o n s i d e r whether t h i s same a p p r o a c h can be used t o d e v e l o p a g e n e r a l i z e d t r e a t m e n t f o r a d s o r b e d n o n i d e a l mixed s u r f a c t a n t m o n o l a y e r s . The p r e f e r r e d form f o r such a model i s t h a t i t be s u i t a b l e ( a t l e a s t i n p r i n c i p l e ) f o r t r e a t i n g m u l t i p l e components and be e x t e n d a b l e t o o t h e r i n t e r f a c e s and p r o p e r t i e s o f i n t e r e s t s u c h as c o n t a c t a n g l e s . E a r l i e r models ( 3 9 18, 27) b a s e d on t h e pseudo-phase s e p a r a t i o n a p p r o a c h and



104

P H E N O M E N A IN M I X E D S U R F A C T A N T S Y S T E M S

r e g u l a r s o l u t i o n a p p r o x i m a t i o n have a more l i m i t e d range o f a p p l i c a t i o n and a r e r e s t r i c t e d t o b i n a r y systems due t o t h e i r f u n c t i o n a l dependence on s u r f a c e t e n s i o n r a t i o s . The form o f the model t o be p r e s e n t e d here i s d i r e c t l y based on p h y s i c a l p a r a m e t e r s and meets the c r i t e r i a f o r g e n e r a l i t y s t a t e d above. Here, the p a r t o f the model which d e s c r i b e s the b e h a v i o r o f the mixed m i c e l l e s p r o v i d e s a b a s i s f o r e s t a b l i s h i n g the c h e m i c a l p o t e n t i a l s o f the i n d i v i d u a l s u r f a c t a n t s p e c i e s i n b u l k s o l u t i o n , and t h e r e b y the e q u i l i b r i u m c h e m i c a l p o t e n t i a l s i n adsorbed mixed monolayers a t the v a r i o u s i n t e r f a c e s w i t h the s o l u t i o n . U s i n g t h i s approach, a model can be d e v e l o p e d by c o n s i d e r i n g the c h e m i c a l p o t e n t i a l s o f the i n d i v i d u a l s u r f a c t a n t components. Here, we c o n s i d e r o n l y the r e g i o n where the adsorbed monolayer i s " s a t u r a t e d " w i t h s u r f a c t a n t ( f o r example, a t o r above the cmc) and where no " b u l k - l i k e " water i s p r e s e n t a t the i n t e r f a c e . Under these c o n d i t i o n s the sum o f the s u r f a c e mole f r a c t i o n s o f s u r f a c t a n t i s assumed t o e q u a l u n i t y . T h i s a p p r o a c h d i v e r g e s from s t a n d a r d t r e a t m e n t s o f a d s o r p t i o n a t i n t e r f a c e s (see r e f 28) i n t h a t the s o l v e n t i s n o t e x p l i c i t l y i n c l u d e d i n the t r e a t m e n t . While the " r e s i d u a l " s o l v e n t a t the i n t e r f a c e can c l e a r l y e f f e c t the s u r f a c e f r e e energy o f the system, we now c o n s i d e r t h e s e e f f e c t s t o be a c c o u n t e d f o r i n the s t a n d a r d c h e m i c a l p o t e n t i a l s a t the s u r f a c e and i n the n o n i d e a l n e t i n t e r a c t i o n parameter i n the mixed pseudo-phase. W i t h t h e s e c o n s i d e r a t i o n s i n mind, the c h e m i c a l p o t e n t i a l o f the i t h f r e e monomeric s u r f a c t a n t component i n s o l u t i o n i s g i v e n by Pi =

μι°

+ RT

In C i

(1)

m

m

where i s a s t a n d a r d s t a t e c h e m i c a l p o t e n t i a l and C ^ the monomer c o n c e n t r a t i o n o f the i t h s p e c i e s (see l i s t o f s y m b o l s ) . F o r pure m i c e l l e s o f the i t h component, a s i m i l a r e x p r e s s i o n

yi

M

o

=

+ RT

In C£

^

r e s u l t s which depends on the cmc o f the pure s u r f a c t a n t , C^. The c h e m i c a l p o t e n t i a l o f the i t h s u r f a c t a n t component i n an a d s o r b e d monolayer o f pure s u r f a c t a n t can be e x p r e s s e d as

y

.s

= .os u

+ .

o s

π

W

(3)

i

π

ω

where y ^ i s a s t a n d a r d s t a t e c h e m i c a l p o t e n t i a l and ^ ^ a f o r c e f i e l d term. C o n v e n i e n t l y , b o t h the s u r f a c e p r e s s u r e and the m o l a r a r e a can be o b t a i n e d e x p e r i m e n t a l l y from i n t e r f a c i a l t e n s i o n measurements ( s u c h as s u r f a c e t e n s i o n s a t the a i r / s o l u t i o n i n t e r f a c e used t o determine the cmc). Here, i s g i v e n by

YH 0-Vi

(

2

and

the m o l a r a r e a

by


4

)

8.

HOLLAND

Nonideal Mixed Monolayer Model

105

Above the c r i t i c a l m i c e l l e c o n c e n t r a t i o n (C^) i n a pure s u r f a c t a n t s o l u t i o n t h e c h e m i c a l p o t e n t i a l o f t h e monomer i s g i v e n by

Mi = and

(6)

+ RT In C i

t h a t o f s u r f a c t a n t s p e c i e s i n t h e a d s o r b e d (pure) monolayer by

y

.s

= .os u

(7)

ma .

+

ïïi

Xa)

max

where ïï i s t h e l i m i t i n g v a l u e o f the s u r f a c e p r e s s u r e above the cmc. A t e q u i l i b r i u m μ^= y ^ and t h e e q u a t i o n s 6 and 7 can be combined t o y i e l d i


s

1

RT RT where the term c o n t a i n i n g t h e c h e m i c a l p o t e n t i a l s i s i n the form o f a bulk-surface d i s t r i b u t i o n c o e f f i c i e n t . I n mixed s u r f a c t a n t systems above the cmc the c h e m i c a l p o t e n t i a l o f the i t h component ( i n the mixed m i c e l l e ) i s g i v e n by

M

Pi

= yi

M

o

(9)

+ RT In f i x i

where f j ^ and x.^ a r e t h e a c t i v i t y c o e f f i c i e n t and mole f r a c t i o n i n the m i c e l l e , r e s p e c t i v e l y . Combining t h i s w i t h e q u a t i o n s 1 and 2 a t e q u i l i b r i u m y i e l d s t h e monomer c o n c e n t r a t i o n as g i v e n i n t h e n o n i d e a l mixed m i c e l l e model (14)

Ci The

chemical

m

=

X

f

i

i

C

(10)

i

p o t e n t i a l o f t h e i t h component i n the monolayer

y

.s

= .os y

+

R

T

l

f s

n

i

x i

s +

π

ω

· .

(^)

now i n c l u d e s a term c o n t a i n i n g i t s a c t i v i t y c o e f f i c i e n t i n t h e mixed monolayer ( f ^ ) a s w e l l as i t s mole f r a c t i o n . As i n t h e case o f mixed m i c e l l e s , the b i n a r y a c t i v i t y c o e f f i c i e n t s based on the r e g u l a r s o l u t i o n a p p r o x i m a t i o n take the form s

ff=

f

s 2

s

S

exp S d - x i )

= exp 3 ( x i ) s

S

(12)

2

(13)

2

where 3 i s a d i m e n s i o n l e s s n e t i n t e r a c t i o n parameter. A t e q u i l i b rium t h e c h e m i c a l p o t e n t i a l s a r e e q u a l and e q u a t i o n s 8 and 9 can be combined t o y i e l d s

s

Ui° -Pi° RT

+

(14)

mJi

=

RT

s

Ui xi


106

P H E N O M E N A IN M I X E D S U R F A C T A N T S Y S T E M S

where π i s t h e t o t a l s u r f a c e p r e s s u r e . Combining e q u a t i o n s 7 and 10 t o e l i m i n a t e t h e i r common term r e s u l t s i n the g e n e r a l i z e d expression


fiX£

(15)

= e

RT

which d i r e c t l y r e l a t e s t h e a c t i v i t y c o e f f i c i e n t s and mole f r a c t i o n s i n t h e mixed m i c e l l a r and monolayer psuedo-phases. Here, t h e i r r a t i o depends on t h e maximum s u r f a c e p r e s s u r e s o f t h e pure s u r f a c t a n t s components, t o t a l s u r f a c e p r e s s u r e , and a r e a s p e r m o l e c u l e a t the i n t e r f a c e i n t h e pure and mixed systems. T h i s a l l o w s compari son between n o n i d e a l i n t e r a c t i o n s i n t h e m i c e l l e and monolayer a s modeled by t h e i r r e s p e c t i v e i n t e r a c t i o n p a r a m e t e r s . Rearranging t h i s expression gives the f o l l o w i n g n o n i d e a l a n a l o g o f B u t l e r ' s e q u a t i o n (29)

RT

Assuming t h e a r e a s p e r m o l e c u l e be f u r t h e r s i m p l i f i e d t o

*- §

In

Î|iîU]

+

TTi

max

Nonideal Mixed Monolayer Model - ACS Symposium Series (ACS

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