Onset of Chaos in the Structure of Microemulsions - ACS Symposium

2 Onset of Chaos in the Structure of Microemulsions E. RUCKENSTEIN Downloaded via YORK UNIV on December 5, 2018 at 23:27:34 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260

A thermodynamic treatment of microemulsions is developed t o explain: (1) the t r a n s i t i o n from a s i n g l e phase microemulsion t o one which coexists with an excess dispersed phase, (2) the t r a n s i t i o n from a two to a three-phase system in which a middle phase microemulsion is in equilibrium with both the excess phases, and (3) the change in structure which occurs near the l a t t e r t r a n s i t i o n point. In addition, the same treatment is employed t o explain observations concerning the i l l - d e f i n e d , fluctuating, i n t e r f a c e s which can a r i s e i n s i d e some s i n g l e phase microemulsions. Because two length scales characterize a microemulsion, two kinds of pressures are defined: the micropressures, defined on the scale of the globules, and the macropressure, defined on a scale which is large compared t o the s i z e of the globules. In contrast to the former pressures, the macropressure is based on a f r e e energy which a l s o accounts, via the entropy of dispersion of the globules in the continuous phase, f o r the c o l l e c t i v e behavior of the globules. The macropressure constitutes the thermodynamic pressure o f the microemulsion and is equal t o the external pressure p. An excess dispersed phase coexi s t i n g with a microemulsion forms when the micropressure p inside the globules becomes equal to p. A third 2

phase, the excess continuous phase, appears when, in addition, the micropressure p in the continuous phase 1

becomes equal to p. The change in structure inside some s i n g l e phase microemulsions as w e l l as near the t r a n s i t i o n from a two t o a three phase system occurs near the point where p = p . This happens because, 2

1

under such conditions, the spherical i n t e r f a c e between the dispersed and continuous media of the microemulsion becomes unstable t o thermal perturbations. These f l u c t u a t i o n s in the shape of the dispersed phase 0097-6156/85/0272-0021$06.00/0 © 1985 American Chemical Society

Shah; Macro- and Microemulsions ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

MACRO- AND MICROEMULSIONS

22

lead t o chaotic v a r i a t i o n s o f i l l - d e f i n e d structures of the microemulsion. The possible existence o f a cascade o f continuous changes from a chaotic micro emulsion to molecular d i s p e r s i o n is a l s o noted. Since Schulman and h i s coworkers discovered microemulsions (1-3), the preferred pathway t o prepare them was t o s t a r t from an emulsion sta b i l i z e d by the adsorption o f surfactant molecules on the surface o f the globules. The a d d i t i o n of a cosurfactant - a medium length a l k y l alcohol - generates an emulsion, containing globules o f almost uni form s i z e , l y i n g between 10^ and 10 A. in contrast t o the conven t i o n a l emulsions, t h i s new kind o f emulsion is, in general, o p t i c a l l y transparent and thermodynamically stable. Because o f the small s i z e of i t s globules, it was c a l l e d a microemulsion. The s p h e r i c a l shape a t t r i b u t e d t o the globules of the dispersed phase is a r e s u l t o f not only our image about the conventional emulsions, but a l s o of numerous experimental investigations (2-7). An a l t e r n a t e pathway, which does not generally use a cosurfac tant (8-10), s t a r t s f r o m m i c e l l a r solutions and involves a nonionic surfactant with ρ ο lyoxy ethylene head group or a double chain i o n i c surfactant, such as Aerosol OT. It is w e l l known that surfactants dissolved in water form, above the c r i t i c a l m i c e l l e concentration, a s u b s t a n t i a l number o f large aggregates (micelles) (11,1 2). Hydro carbon molecules, though i n s o l u b l e in water, can be s o l u b i l i z e d e i ther among the hydrocarbon chains o f the m i c e l l e s (13-15) or f o r some surfactants, such as those with ρ ο lyoxy ethylene head group, the solu b i l i z e d molecules can form a core surrounded by a l a y e r of surfactant (a microemulsion). M i c e l l a r aggregates are s p h e r i c a l f o r small s i z e s and c y l i n d r i c a l f o r large s i z e s (11). When the s o l u b i l i z a t i o n occurs among the hydrocarbon chains, s p h e r i c a l o r c y l i n d r i c a l shapes are a l s o expected t o occur (14,1 5), depending upon the s i z e o f the aggregate. When the aggregate consists o f a core o f hydrocarbon mole cules protected by a l a y e r of surfactant, i n t u i t i o n suggests that a s p h e r i c a l structure w i l l r e s u l t f o r aggregates of any s i z e (14,15). However, since the i n t e r f a c i a l tension at the surface o f the globules is very low, the entropie freedom which a non-rigid, non-spherical shape can provide may overcome the e f f e c t o f the increase in area. As a r e s u l t , non-spherical globules could be preferred thermodynamically. Surfactants which are dissolved in oil form small, non-spherical aggregates (16,17). However, the presence o f water favors the forma t i o n o f "swollen micelles", containing a core of s o l u b i l i z e d water molecules. Again, i n t u i t i o n suggests a s p h e r i c a l shape f o r these globules. For completeness, l e t us note that i f the volume f r a c t i o n of the dispersed phase becomes s u f f i c i e n t l y large, it is expected that the i n t e r a c t i o n s between the globules w i l l a f f e c t t h e i r shape. As these i n t e r a c t i o n s become greater, a p e r c o l a t i o n threshold occurs. The e l e c t r i c a l conductivity o f the system increases steeply a t the threshold because o f the transient interconnections between an i n f i n i t e number o f globules o f water. More recent experiments have indicated that the structure o f the d i s p e r s i o n could be much more complex. The f i r s t o f these observa t i o n s r e s u l t e d from a study o f the phase behavior o f microemulsions. 3


2. RUCKENSTEIN

Chaosinthe Structure of Microemulsions

23

This is best i l l u s t r a t e d by what happens when the amount of s a l t is increased, f o r f i x e d amounts o f oil, water, surfactant and cosurfactant (18-20). For r e l a t i v e l y low amounts o f s a l t , an oil in water microemulsion coexists with excess oil phase; a t s u f f i c i e n t l y high s a l t content, a water in oil microemulsion is in equilibrium with excess water phase, while a t intermediate s a l i n i t i e s , a (middle phase) microemulsion coexists with both water and oil excess phases. Of course, the excess phases contain some surfactant, cosurfactant and even water or oil a t concentrations determined by the equilibrium between the phases. In the present context, it is important t o note that, in general, the microemulsion contains spherical globules whose s i z e d i s t r i b u t i o n is f a i r l y uniform when it coexists with a s i n g l e excess phase, but that i t s structure becomes very complex as soon as it coexists in equilibrium with both excess phases. Very recently, a novel F o u r i e r transform NMR method was employed by Lindman, e t a l . (21) t o obtain multicomponent s e l f - d i f f u s i o n data f o r some s i n g l e phase microemulsion systems. Because o f the large values obtained f o r the s e l f - d i f f u s i o n c o e f f i c i e n t s o f water, hydrocarbon, and alcohol, over a wide range of concentrations, the authors concluded that there are no extended, well-defined structures in these systems. In other words, the i n t e r f a c e s which separate the hydrophobic from the hydrophilic regions appear t o open up and reform at a short time scale. The scope o f the present paper is t o show that the thermodynamic theory developed by the author (22,23) can e x p l a i n the change in structure which occurs near the point of t r a n s i t i o n from two t o three phases as w e l l as in some s i n g l e phase microemulsions. In essence, the thermodynamic considerations that follow demonstrate the existence of a t r a n s i t i o n point in the v i c i n i t y of which the s p h e r i c a l shape of the globules is no longer stable and is replaced by a c h a o t i c a l l y f l u c t u a t i n g i n t e r f a c e . For t h i s reason, it is appropriate t o c a l l t h i s t r a n s i t i o n the spherical t o chaotic t r a n s i t i o n . The next s e c t i o n of the paper contains a q u a l i t a t i v e explanation of the above phenomena. The corresponding thermodynamic equations are then der i v e d and an explanation concerning the o r i g i n of the middle phase microemulsion as w e l l as the i n t e r p r e t a t i o n of the NMR experiments f o r some s i n g l e phase microemulsions f o l l o w . A d i s c u s s i o n regarding the " t r a n s i t i o n " from macroscopic t o molecular d i s p e r s i o n concludes the paper.

Q u a l i t a t i v e Considerations The d i s p e r s i o n of one phase i n t o a second phase in the form of globu l e s leads t o an increase in the entropy o f the system and r e s u l t s in the adsorption o f surfactant and cosurfactant on the large i n t e r f a c i a l area thus created. This adsorption decreases the i n t e r f a c i a l tension from about 50 dyne/cm, c h a r a c t e r i s t i c of a w a t e r - o i l i n t e r face devoid of surfactants, t o some very low p o s i t i v e value. In add i t i o n , the concentrations of surfactant and cosurfactant in the continuous and dispersed phases are decreased as a r e s u l t of the adsorption, thereby reducing t h e i r chemical p o t e n t i a l s . This d i l u t i o n of the bulk phases leads t o a negative f r e e energy change, which we c a l l the d i l u t i o n e f f e c t . Dispersions that are thermodynamically


24


stable are generated when the negative f r e e energy change due t o the d i l u t i o n e f f e c t and t o the entropy of d i s p e r s i o n overcomes the posi t i v e product of the low i n t e r f a c i a l tension and the large i n t e r f a c i a l area produced. This explains the thermodynamic s t a b i l i t y of micro emulsions (24). Let us now consider that the microemulsion contains s p h e r i c a l globules. These globules are, however, macroscopic bodies. Conse quently, two macroscopic length scales characterize a microemulsion. One of them is the scale of the globules, while the other, which is much l a r g e r than that of the globules, is the scale of the microemul sion. The thermodynamic pressure is defined a t the s c a l e of the microemulsion, on the b a s i s o f a f r e e energy which includes the "macroscopic" behaviour r e f l e c t e d in the entropy of d i s p e r s i o n of the globules in the continuous phase. In addition, one can, however, de f i n e pressures a t the scale of the globules. Let us c a l l the pressure p^ i n s i d e the globule and the pressure p^ near the globule in the con tinuous phase micropressures, t o s t r e s s the f a c t that they are de f i n e d on the s c a l e of the globules and t o contrast them to the macroit he rmodynamic) pressure which is defined on the s c a l e of the e n t i r e microemulsion. A p a r t i c u l a r globule which has the pressure ρ inside 2 f e e l s outside, in i t s v i c i n i t y , the micropressure p^. Only a "macro scopic" part of the microemulsion, which of course should be large compared t o the s i z e of the globules, f e e l s the thermodynamic pres sure. In other words, the micropressures are defined on the b a s i s of a f r e e energy which does not include the entropy of dispersion o f the globules in the continuous phase (an e f f e c t which manifests it s e l f a t a s c a l e of the order o f the microemulsion, which is large compared to the s i z e of the globules). Because of the condition of mechanical equilibrium, the thermodynamic pressure equals the pres sure ρ of the environment (external pressure). It is important t o emphasize the ρ and ρ are r e a l pressures a t the s c a l e of the glob2 1 u l e s , and that the thermodynamic pressure is a r e a l pressure in a volume which contains a l a r g e number of globules. Let us consider one globule a t the i n t e r f a c e between microemul s i o n and environment. As long as the micropressure ρ < ρ, the 2 globule w i l l stay in the microemulsion and no excess dispersed phase w i l l appear, because the chemical p o t e n t i a l a t the pressure p^ is smaller than that a t the pressure p. The mechanical equilibrium con d i t i o n between microemulsion and environment is s t i l l s a t i s f i e d be cause the macro (thermodynamic) pressure is equal t o the external pressure. For p^ > ρ the globules w i l l disappear from the continu ous phase, i . e . , the microemulsion w i l l give way t o separate oil and water phases. An excess dispersed phase in equilibrium with the microemulsion w i l l appear when ρ - p. Similar considerations show 2 that a t h i r d phase, the excess continuous phase, appears when ρ - ρ·


2.

RUCKENSTEIN

25


Consequently, a three-phase system composed of both excess phases and a microemulsion w i l l form when p^ • p^ - p. However, the equality p^

β

p^ is not compatible with the existence o f s p h e r i c a l globules,

because, near such a point, the i n t e r f a c e becomes unstable t o thermal perturbations and, therefore, it f l u c t u a t e s . Thus, the change in structure observed experimentally occurs near the t r a n s i t i o n point from a two t o a three-phase system, as a r e s u l t o f the equality o f the micropressures a t that point. Of course, the micropressures p^ and p^ can become equal without any phase separation ( i . e . , without t h e i r common value being equal t o p). For t h i s reason, f l u c t u a t i o n s of the s p h e r i c a l i n t e r f a c e are a l s o expected t o a r i s e in some s i n g l e phase microemulsions. This explains the NMR experiments of Lindman, et a l . (21> The above simple considerations explain the o r i g i n of the middle phase, the change in structure associated with i t s occurrence, as w e l l as the f l u c t u a t i o n s o f the i n t e r f a c e between the continuous and dispersed medium which a r i s e in some s i n g l e phase microemulsions. While it is d i f f i c u l t t o o b t a i n d e t a i l e d quantitative information on the above behaviour, the thermodynamic equations derived in the f o l lowing s e c t i o n provide a framework f o r further t h e o r e t i c a l develop ment as w e l l as some a d d i t i o n a l i n s i g h t concerning the micropressures and various p h y s i c a l q u a n t i t i e s involved.

Basic Thermodynamic Equations Let us assume that the microemulsion contains s p h e r i c a l globules o f a s i n g l e s i z e . For given numbers o f molecules o f each species, tem perature and external pressure, we consider an ensemble o f systems in which the r a d i i and volume f r a c t i o n s o f the globules can take ar b i t r a r y values. Because the equilibrium state o f the system is com p l e t e l y determined by the number of molecules o f each species, the temperature and external pressure, the actual values o f the radius and volume f r a c t i o n w i l l r e s u l t from the condition that the f r e e en ergy o f the system be a minimum. We s t a r t with the observation that the d i s p e r s i o n o f the glob u l e s in the continuous phase is accompanied by an increaseinthe entropy o f the system and denote by Af the corresponding f r e e energy change per u n i t volume o f microemulsion. However, l e t us consider, f o r the time being, a "frozen" system o f globules i n s i d e the contin uous phase, ignoring Af. At constant temperature, the v a r i a t i o n o f the Helmholtz f r e e energy per u n i t volume, f , o f the frozen micro emulsion can be written, i f the actual physical surface o f the glob u l e s is used as the (Gibbs) d i v i d i n g surface, as follows (25,22,23); 0

df ο

- γ dA + Cd(l/r) + Σμ dn - ρ d* - ρ d ( l - Φ) 1 1 2 1

(1)

where γ is the i n t e r f a c i a l tension, C is the bending s t r e s s due t o the curvature 1/r, A is the i n t e r f a c i a l area per u n i t volume o f micro emulsion, n^ is the number o f molecules o f species i per u n i t volume,


26


μ^ is the electrochemical p o t e n t i a l of species i , p^ and p^ are the micropressures i n s i d e the globules and in the continuous phase, re s p e c t i v e l y , and φ is the volume f r a c t i o n of the dispersed phase. Of course, the radius r , the area A, the volume f r a c t i o n φ, the i n t e r f a c i a l tension γ and the bending s t r e s s C correspond to the selected (Gibbs) d i v i d i n g surface. The area A and the radius r of the glob u l e s are r e l a t e d via the expression A = 3φ/Γ

(2)

We note that Equation 1 contains the micropressures

p^ and p^,

since they represent the pressures in the continuous and dispersed media of the frozen state, the c o l l e c t i v e behaviour expressed via Af being Ignored f o r the frozen state. One can remark that the i n t e r f a c i a l tensionΎ , which is defined by a v a r i a t i o n in the frozen f r e e energy with area at constant φ, n^ and T, includes a l s o those i n t e r a c t i o n s between globules such as the van der Waals, double layer and hydration force i n t e r a c t i o n s , which change with surface area A. However, a change in A at constant φ changes a l s o the shortest distance, 2h, between the surfaces of two neighboring s p h e r i c a l globules. Thus, the v i r t u a l change used to define γ changes the s p a t i a l d i s t r i b u t i o n of the globules and not j u s t t h e i r surface area. Therefore, γ includes, in a d d i t i o n to the e f f e c t of the above i n t e r a c t i o n s at a given distance 2h, the e f f e c t of the r e v e r s i b l e work done against these forces, as h changes. Con s i d e r i n g the f o r c e τ per u n i t area to be p o s i t i v e f o r repulsion, the v a r i a t i o n 2dh produces, per u n i t volume of microemulsion, the work Axdh. A more f a m i l i a r i n t e r f a c i a l tensionY^, from which the e f f e c t of the v a r i a t i o n of h is eliminated, can be therefore defined by γ dA = γ dA h

- Ατ 9 A

(3)

dA Φ,ΐΗ,Τ

Since the equations w r i t t e n in terms of γ are more compact than those in terms of γ^, the treatment which follows uses Equation 1 as the s t a r t i n g point. The f r e e energy per u n i t volume of microemulsion is given by the sum f - £o + Af

(4)

which, when combined with Equation 1, leads to: df = γ dA + Cd(l/r) + Σμ dn

- ρ άφ - ρ d ( l - φ) + dAf

(5)

While a l l the v a r i a b l e s r , η^, φ, Τ and ρ are necessary to s p e c i f y an a r b i t r a r y state, the equilibrium s t a t e of a microemulsion is com p l e t e l y determined by η^, Τ and p. The values of r and φ w i l l there f o r e emerge from the condition that the microemulsion be in i n t e r n a l


2. RUCKENSTEIN

27


equilibrium, i . e . , f be a minimum with respect t o r and φ (or A and φ ) . This leads t o the equations:

- - (|τ·)

y

φ

- C ( ^ ^

(constant n

+

and T)

t

(6)

and P

2

- C|y f 9 A

" ?1

+ C

K

(constant n

t

and T)

For spherical globules, A and r a r e r e l a t e d via Equation 2. Equations 6 and 7 become:

Then

( 8 )

3Γ

~ W and, since

(7)

f

3Af^

-

3Af-,

/dr-, . 3Af-, +

r

f

It is i n s t r u c t i v e t o write Equation 9 in another equivalent form. 4 ο

At

J

constant m, where m (= φ/y i r r ) is the number of globules per u n i t volume of microemulsion, (3Af>

_ 3φ 3Af>>

3Af>>

r

r

Combining t h i s equation with Equations 8 and 9, one obtains: 2γ C . r 3Af>> 2 " P i — " 3φ? 3f(3r-Jm r

p

=

/ Λ

m

( 1 0 )

+

Equation 10 reveals more obviously that Equation 9 constitutes a gen e r a l i z e d Laplace equation. The mechanical equilibrium c o n d i t i o n between microemulsion and the environment (which is a t the constant pressure p) provides a second r e l a t i o n between ρ and ρ · Indeed, the v a r i a t i o n of the 2 1 Helmholtz f r e e energy F o f the e n t i r e microemulsion can be w r i t t e n as

dF =ïd(AV) + VCd(l/r) + E y ^ i - ρ^(νφ) - ρ^(ν(1 - φ))

+d(VAf) where V is the volume of the microemulsion and is molecules of species i in the e n t i r e microemulsion.

the number of Considering a



28

v a r i a t i o n dV of the volume a t constant Τ, ρ and N^, the mechanical equilibrium c o n d i t i o n with the environment y i e l d s the equation: γ d(AV) + VCd(l/r) - ρ d(V) - ρ d(V(l - φ)) + d(VAf) 2 1 - pdV = 0

(11)

e

where dV = - dV is the v a r i a t i o n of the volume of the environment. Combining Equation 11 with Equations 8 and 9, one obtains: e

§*-Ύ

- ( P " Ρχ)Φ + (Ρ 2

+ Af - 0

(12)

The system of Equations 8, 9 and 12, lead t o the following expressions f o r ρ and ρ : ^ • - - " « - « ( ^ . { © , - ΐ ,

Pj - ρ • Af - φ (||£-)

(14)

and r

Equations 13 and 14 r e l a t e the micropressures p^ and p^ to the ex t e r n a l pressure p,C and t o Af and i t s d e r i v a t i v e s . It is obvious that the mechanical equilibrium c o n d i t i o n requires the macro (thermodynamic) pressure of the microemulsion t o be equal to p. Equations 13 and 14 r e v e a l that the macropressure is equal top or ρ plus terms which a r i s e as a r e s u l t of the c o l l e c t i v e 2 1 behavior of the globules r e f l e c t e d in Af and a l s o due t o the curvature e f f e c t .

The O r i g i n of the Middle Phase Microemulsion and of i t s Structure 3f The chemical p o t e n t i a l in the microemulsion phase is defined as g-j^ at constant Τ, Α, φ and nj (with j * i ) . Because the f r e e energy Af due t o the entropy of d i s p e r s i o n of the globules in the continuous phase can be considered a f u n c t i o n of only r and φ and independent 3f of m (2 6,2 7), it follows that τ is equal t o the chemical p o t e n t i a l — à nj Vi in the frozen state. Assuming the concentrations of various components t o be the same in the globules and the excess dispersed phase, the equality of the chemical p o t e n t i a l s is equivalent t o the equality o f the pressures. The chemical p o t e n t i a l s in the dispersed phase are expressed in Eq. (5) a t the micropressure ρ . Since in the excess 2 dispersed phase the pressure is equal t o the external pressure, an excess dispersed phase forms when p

2

-P


(15)

2.

RUCKENSTEIN

29

Chaos in the Structure of Microemulsions

which, when introduced in Equation 13, provides the expression r3Af>>

C

. r 9Af r

(16)

Equations 8 and 16 provide the basic thermodynamic equations which can p r e d i c t the dependence on s a l i n i t y , of the equilibrium radius r and the volume f r a c t i o n φ at the t r a n s i t i o n between the region in which a microemulsion phase forms alone and that in which it coexists with an excess dispersed phase. Any a d d i t i o n to the system of excess dispersed phase having the same composition as the globules w i l l change neither φ nor r in the microemulsion, as soon as the t r a n s i t i o n point is reached. To carry out such c a l c u l a t i o n s , e x p l i c i t ex pressions are needed f o r the i n t e r f a c i a l tension γ as a function of the concentrations of surfactant and cosurfactant in the continuous phase, of s a l i n i t y and radius r , as w e l l as expressions f o r C and f o r the f r e e energy Af. The i n t e r f a c i a l tension depends on the radius f o r the following two reasons: If the radius were increased at con stant φ, the surface area becomes smaller; t h i s decreases the t o t a l amount of surfactant and cosurfactant adsorbed, but, because the sys tem is closed the concentrations in the bulk become l a r g e r and the amount adsorbed per u n i t area increases. Thus, the i n t e r f a c i a l ten s i o n decreases. In a d d i t i o n to the above mass conservation e f f e c t , there is a l s o a curvature e f f e c t , due to the following r e l a t i o n which e x i s t s between γ and C (the generalized Gibbs adsorption equation (25,23)): 3*Y

C

γ£ = - 3^(constant Mi and T) It is d i f f i c u l t to derive an equation f o r γ , p a r t i c u l a r l y a t high elec t r o l y t e concentrations. However, f o r a planar i n t e r f a c e , with only a s i n g l e surfactant, such an equation was recently established (28). I t is a l s o d i f f i c u l t to e s t a b l i s h a r e l a t i o n f o r C. As f o r A f , ex pressions have already been derived, e i t h e r on the b a s i s of a l a t t i c e model (26), or on the b a s i s of the Carnahan-S t a r l i n g approximation f o r hard spheres (2 7). These expressions could be introduced in Equa t i o n s (8) and (16) to r e l a t e r and φ to γ and C, or perhaps even more meaningfully, to r e l a t e γ and C to r and φ. S i m i l a r l y , from the e q u a l i t y of the chemical p o t e n t i a l s in the continuous and excess continuous phases one concludes that a t h i r d phase, the excess continuous phase, appears, when, in a d d i t i o n to ρ = ρ, we a l s o have

Ρ

- Ρ

(17)

which, when combined with Equation 14, leads to A

f

- * (Ir3

=

0

( 1 8 )

r

A change in structure is expected to occur near the t r a n s i t i o n to the three phase system, since the e q u a l i t y p^ = p^ = ρ is not



30

compatible w i t h spherical globules. Large f l u c t u a t i o n s of the i n t e r face between the two media of the microemulsion are expected to occur in the v i c i n i t y of such a point. This behaviour, which is sim i l a r to that occurring near a c r i t i c a l point, explains the r e s u l t s of the l i g h t s c a t t e r i n g experiments of Cazabat et a l . obtained near the t r a n s i t i o n to the three-phase system and in the middle phase region. These authors note the increase in t u r b i d i t y , the decrease of the d i f f u s i o n c o e f f i c i e n t , the large thicknesses of the two i n t e r faces, as w e l l as the very small values of the i n t e r f a c i a l tension between the middle phase and each of the excess phase, near the tran s i t i o n point. It is a l s o of i n t e r e s t to note that the i n t e r f a c i a l tensions between the microemulsion and each of the excess phases could be represented by expressions which are v a l i d near a c r i t i c a l point (29). While the above observations are t y p i c a l l y v a l i d in the v i c i n i t y of a c r i t i c a l point, they are, in the case of microemulsions, a r e s u l t of the f l u c t u a t i o n s of the spherical i n t e r f a c e , which occur in the v i c i n i t y of the point where ρ • ρ • p. The above thermody2 1 namic equations are no longer v a l i d in t h e i r i n i t i a l form f o r the middle phase microemulsion, because, in t h i s case, we no longer have s p h e r i c a l globules of only one s i z e . A d d i t i o n a l l y , chaotic breakup and coalescence are a l s o probably taking place. To transform the above equations i n t o a p r e d i c t i v e t o o l is not an easy task, f o r reasons already outlined. They constitute, however, a thermodynamic framework on the basis of which further development could follow. Discussion of The NMR Experiments in Single Phase Microemulsions

As already noted, a p a r t i c u l a r globule, which has the pressure p^ in side, f e e l s outside, in i t s v i c i n i t y , the micropressure p^.

The con

d i t i o n of mechanical equilibrium of the spherical i n t e r f a c e of the globules requires p^ > p^.

However, it is important to r e a l i z e that,

in contrast to the case of a l i q u i d droplet surrounded by i t s vapors (which is treated in Reference 25), t h i s i n e q u a l i t y does not require r^~ " 3^r~

t

0

^

e

a

P

o s i t i v e

quantity, since the a d d i t i o n a l

term^(|^-)

which appears in Equation 10 is always a p o s i t i v e quantity. Indeed, the increased volume exclusion (which a r i s e s when the radius r in creases at constant number m of globules) decreases the disorder and hence the entropy of dispersion of the globules in the continuous phase. As a r e s u l t , the corresponding f r e e energy Af increases with increasing radius, at constant m. As the q u a l i t a t i v e discussion of a previous s e c t i o n demonstrates, one may a l s o note that ρ < ρ in s i n g l e phase microemulsions and ρ^ excess dispersed phase.

β

ρ when a microemulsions coexists with an

Therefore, in order to examine the mechanical


2. RUCKENSTEIN


31

s t a b i l i t y o f the s p h e r i c a l i n t e r f a c e o f the globules, it is not appro p r i a t e t o compare p^ with the external pressure as one might be tempted t o do* The s p h e r i c a l shape of the globules is stable t o thermal pertur bations as long as ρ > ρ and becomes unstable in the v i c i n i t y o f 2 1 the point where p^ • p^.

At t h i s point, Equations 9 and 10 lead t o

the expressions:

(Ml) ^3φ

>

+ Έ

+ L(*ÈL φ ^3r \

_Ç_=2L._C_ " r 3*r

+

+

r_fi*L) l

3* 3r

=

j

(19)

0 0

e

The i n s t a b i l i t y of the s p h e r i c a l shape occurs near the point where :r?

—

j

Onset of Chaos in the Structure of Microemulsions - ACS Symposium

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