micelles, vesicles and micro emulsions

8/2/2019 Micelles, Vesicles and Micro Emulsions

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J. Chem. SOC.,araday Trans. 2, 1 9 8 1 , 7 7 , 601-629

Micelles, Vesicles and MicroemulsionsB Y D. JOHNMITCHELLAN D BARRYw. INHAM

Department of Applied Mathematics, Institute of Advanced Studies, ResearchSchool of Physical Sciences, The Australian National University, Canberra, A.C.T.2600, AustraliaReceived 25th June, 1980

A theory of self-assem bly of surfactant molecules into micelles and bilayers is critically exam ined andextend ed to include vesicles and microemulsions. Th e notion of hydrophilic-lipophilic balance isquantified. Th e theory gives a unified account of type, size and shape of the aggregates which form undervarious condition s. Observed trends due to change in salt concentr ation, temperature and oil type, anddue to the addition of cosurfactants, are correlated and em erge from a simple global framework.1. I N T R O D U C T I O N

Until a few years ago the possibility that all observations on association colloidsystems could ultimately be handled by a single theoretical framework seemedremote. It became less so following attemptslY2o extend the ideas of Tanford3 andothers4 on dilute micellar aggregates to larger surf actant associations like cylindricalmicelles, vesicles and bilayers. The main point of departure lay in quantifying thepart played by molecular geometry (packing) in determining allowed structures. Itwas an old idea5 which had been allowed to lie fallow. And it worked. Theory doesappear to be on the right track. While there are gaps, parts of the jigsaw puzzle havebeen filled in more or less satisfactorily for dilute surfactant solutions. Certainlymany of the physical properties of micelles and vesicles like size, shape, c.m.c. andpolydispersity appear to be accessible without a detailed knowledge of the complexintermolecular forces involved.Our purpose here is twofold: (1)To attempt to define better and explore some ofthe basic assumptions which underlie ideas presently extant. (2) To see how theseideas might be extended to include multicomponent systems (microemulsions).From a pragmatic point of view, one main aim of studies in the subject must surelybe: to elucidate the phase diagrams of water-surfactant (and cosurfactant)-hydro-carbon mixtures; in particular to identify which structures form, when and why; andas a corollary: how to maximise solubilisation of oil in water, or water in oil, with aminimum surfactant (cosurfactant) concentration.This aim is ambitious, and the problem of such complexity that, to paraphrase andborrow a remark made by Stillinger:6 it is essential to maintain a respectablebalance between the sterile intricacy of formal theory and the seductive simplicity ofpoetic explanation. Before beginning our study it may be useful to expand thisdictum. In attempting to make a theory there are two extreme approaches. Afundamental treatment using statistical mechanics which takes into account complexsurfactant molecule interactions in water is possible in principle. However, even thehydrophobic interaction between two small molecules in water is still a matter ofdispute. Further, the simplest prototype for aggregation, the probem of nucleation(and consequent phase transition) in a van der Waals gas, is an open subject.Moreover, the high road via statistical mechanics is necessarily so complicated thatphysical insight tends to be wholly obscured.

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602 M I C E L L E S , V E S I C L E S A N D M I C RO E M UL S I ON SAt the other end of the spectrum one can avoid detailed models as far as possibleand search for a unified thermodynamic picture like that of T a n f ~ r d , ~nd of

Israelachvili et a1.l for micelle and bilayer formation. But thermodynamics istautological and can go so far. At a certain point some details of molecularinteraction must be invoked. Our guiding principle in attempting to steer a middlecourse between these two extremes will be that these details must be minimal.Otherwise, with too many (unknown) parameters, theory tends to become anexercise in curve fitting, a numerical game which loses predictive capacity andcredibility.An immediate consequence is that language must be used with some care. Theproblem is here doubly compounded and confounded by the fact that words likeamphiphile, hydrophobic, hydrophilic, lipophilic, aggregate, micelle, are eitheranthropomorphic in origin (disguising their complexity) or intuitive, ill-defined, andare so familiar that we tend not to question their meaning. This question ofdefinitions will plague us throughout and is unavoidable. A result is that our essayhas elements of schizophrenia, with necessary appeal to formal statistical mechanics(relegated to an appendix) interspersed with (hopefully) an occasional insightgleaned from intuition and simple models.

2. T H E R M O D Y N A M I C S OF D IL -U TE S O L U T I O N S O FS U R F A C T A N T SDilute micellar systems have been the subject of an extensive li terature which wedo not wish to recapitulate in detail. Given certain reasonable assumptions they

are broadly speaking understood. However, conflicting opinion on questions like thestability of vesicles and the delicacy of experiments on microemulsions, to name twoareas of many, will demand re-examination of foundations as we proceed.For the moment it suffices to assert that a dilute solution of surfactant moleculescan be considered to consist of solvent plus monomers, dimers, trimers, . . .and largerallowed aggregates (micelles, vesicles, liposomes, . . .). The concentration is assumedto be so low that aggregates can be considered to be non-interacting. The dis-tribution of aggregates is then determined by the law of mass action, eqn ( l ) ,kT XNP & + In (N) p ? +kT In X l ,

where the chemical potential of an aggregate of size N has been written asNp&+ k T In ( X N / N )nd XN is the concentration (mole fraction, volume fraction,. . .) of surfactant molecules in the N-aggregate. The theory also allows for thepossible formation of infinite aggregates, i .e . separate phases, a problem which wedefer. The glib assertion eqn (1) epresents a beginning to a chemist and an essentialstumbling block to a physicist who can go no further without questioning origins. Ifaggregates of a given N were distinct, identical well-defined chemical species therewould be no problem, apart from the vexed question of concentration units. Theyare not: even within a given N-aggregate, if such can be defined, there exist aninfinite diversity of conceivable shapes or configurations which the association ofsurfactant molecules could take up. Implicit in eqn (1) s the understanding that forany N a shape of minimum energy exists and is overwhelmingly more probable thanits fellows. These problems are addressed in the Appendix. If the arguments thereadvanced are accepted we return to eqn (1)which can be rewritten in the formXN- = X y exp [ - N ( p & - - p ? ) / k T ] .N (2)

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D . J . M I T C H E L L A N D B . W . N I N H A M 603All aggregates can occur at any concentration, albeit with infinitesimal probabilityeven below the critical micelle concentration (c.m.c.). Above the c.m.c., defined byXI z:, XN,X I ncreases slowly with concentration. It is shown in ref. (1 ) that ifp& is sharply distributed about some N then the distribution of aggregates peaks at avalue of N just less than the N with minimum p& and is also sharply distributed.Otherwise pronounced polydisperity may occur (e .g .for long, cylindrical micelles).Thus reduced to bare bones, it can be seen that the use of the word theory isdubious. We have simply characterized the observation of micelles, and have shownvia the Appendix that the law of mass action is an appropriate vehicle for thischaracterisation.The entropic term ( k T / N ) n (XN/N) n eqn (1)has considerable nuisance value.For that reason it will usualIy be helpful in dealing with finite aggregates to adopt asimpler basis fo r our subsequent development. This is the so-called pseudophaseapproximation wherein one drops the entropic term. In the pseudophase approxi-mation n o micelles occur below the c.m.c. (This is now the value XI of monomerconcentration for which p& = p ; +kT ln Xl.) Above this c.m.c. all additionalsurfactant molecules form micelles or whatever aggregate has the minimum p &. N oother aggregates form until activity coefficients, i .e. interactions between aggregates,become significant. Keeping in mind possible complications due to phase transitionsand interactions the strategy is then to comp,-ire the chemical potentials of differentaggregates to see which has the minimum p&.

T H E F O R M O F p&In essence the simplest version of current theory assumes a form

p& = p: +p k +p: +packing term= bulk term +surface term + curvature term +packing.Its decomposition is as follows (cf .fig. 1).

(3)

waterFIG.1.--Schematic representation of mo del spherical micelle of radius R.

Bulk Term :This is a constant term, the same for all aggregates, which measuresthe hydrophobic free energy of removing hydrocarbod tails from water into anassumed oil-like phase made up of all tails which form the micelle interior. Theinterior is assumed to be fluid in estimating this free energy transfer.Surface Ter m :This includes a ter mt ya to allow for the fact that hydrophobic tailsstill have some contact with water, where a is the area per surfactant molecule, and yis an interfacial tension.

t It is a trivial decoration to exc lud e that fraction of the o il-water interface taken up by head groups.

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604 MICELLES, V ES IC LES A N D M I C R O E M U LS I O NSOpposing this attractive energy is a term to account for repulsive head-groupinteractions. These interactions can be due to steric repulsion, hydration, electro-static and other forces. They are not yet quantified. If electrostatic in origin the

simplest phenomenological description would give a term a /a . The total surfacecontribution would then be writtenp R = y ( a +2) (4)

which takes its minimum value 2 y a o at an optimal area a. per head group. This formfor the repulsive contributions to p s cannot be taken literally, and its precise formis immaterial, Any mathematical description of the surface contributions whichrecognises that several competing forces will lead to an optimal area a. per headgroup will lead to the same conclusions.Curvature Term : If as an aid to analysing data on ionic micellar solutions we wereto persist with an electrostatic model for head-group repulsions, then for a curved,rather than planar, surface the term a in y a i / a would be replaced by a(1+ D / R )where R is the mean radius of curvature and D the Debye length. An alternativeway of visualising a mechanism for curvature effects is to imagine that the centre ofaction for head-group repulsion is displaced a distance D from the oil-waterinterface. Then a would be replaced by a (1+ D / R ) 2 .We choose the first form againonly to have some plausible mathematical realisation of the undoubted existence ofcurvature energy.Packing Term :The assumption that the interior of an aggregate is fluid-like and isto a first approximation incompressible has an immediate consequence, provided weadmit that aggregates can obtain no holes. [The occurrence of an interior vacuum ofwater-filled region inside the (oil-like) interior of an aggregate would result in a largeunfavourable increase in free energy, which possibility is excluded fromconsideration].? This can be taken into account if we assume p k = 00, hence apacking criterion is violated. For spherical and cylindrical micelles this criterion isR < , where I , is a critical tail-length which in ref. (1)was taken to be ca. 80-90% ofthe fully extended chain length for bilayers. The packing criterion is clearly anextreme, albeit useful, oversimplification. We shall see later that it can, and indeedmust be relaxed. The melding of the two notions, of a fluid-like interior for themicelle and of packing, is at first sight contradictory. However, the two notionscan be shown to be compatible in a first-order theory.8In addition to the contributions above to p & there will be others with increasingconcentration due to interactions.3. G E O M E T R I C A L C O N S I D ER A T I O N S A N D Z E R O T H - O R D E RT H E O R Y

Possible candidates for aggregates can now be examined. For surfactant-watersystems these are spherical micelles, non-spherical micelles (globular and cylindri-cal), vesicles, liposomes, bilayers, and for oil-surfactant-water systems sphericaldrops, normal or inverted (w5ter dispersed oroil dispersed, respectively). Weassume that the surfactant molecule can be divided unambiguously into a head groupand tail and that the partial volume v of the tail is prescribed so that for an aggregateof number N the tail region volume is N v .The location of the interface which defines the radius R is a difficult question withdifferent conventions adopted by different a~thors .~However if the theory isnot to

t Note added in proof: Experimental observa tions which hav e sometimes been interpreted as implyingwater penetration in the hydrophobic core of micelles are completely ex plicable in terms of the samplingof the surface groups in the chains (D.W. R. Gruen, to be published).

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D. . M I T C H E L L A N D B. W . N I N H A M 605depend o n specific details, the convention must also not be critical. Take theinterface to be the surface which bounds the volume occupied by hydrocarbon tails.Then for spherical micelles (fig. 1)we have $n-R = N v and a head-group area definedby 4mR2= N a . The packing criterion imposes the condition that the average chainlength is R s I,.By zeroth-order theory we mean one in which a = a o , the optimum area for abilayer, independent of the size and shape of the micelle. (Head-group curvature isassumed to be a small perturbation.) Because the zeroth-order theory prescribesa = a o , the dimensions of the spherical micelle are completely determined (R =3 v / a o ) , i .e. R > ,. Then if v / a o l ,> 3, the optimum-energy spherical micelle isdisallowed by packing because then R > ,. On the other hand, for R s , sphericalmicelles allowed by packing are energetically disfavoured (because a > ao). Inhigher-order theories a will be determined by optimisation of p & and the optimumspherical micelle (R d ,) will be compared with other shaped aggregates.

Problems of packing are extremely complex for shapes of low symmetry. Forshapes of high symmetry the packing criterion is easy to visualise. In other cases thesimplest condition is that no interior point can be less than a distance I , from thesurface. This criterion leads to the conclusion that only oblate shapes shouldoccur.l*lo I n practice this is not so, and large prolate or rod-shaped micelles areobserved.1928To overcome this difficulty in ref. (1) he authors imposed a plausible restricted(local) packing criterion. As there shown, the pseudophase approximation breaksdown, and large polydisperse cylindrical micelles form. Their size distributiondepends on concentration. So as not to distract from our main theme it will besufficient to treat non-spherical micelles as if they were infinite cylinders. Forcylinders R = 2 v / a o so that packing requires vlaol , 1. I n this region we have to searchfor different allowed structures. These could be, e .g . , vesicles, liposomes or multi-lamellar aggregates. For the moment consider only vesicles and single bilayers. Indiscriminating between these, the head-group curvature again favours the smallestaggregates as does entropy, i.e. the smallest vesicle allowed by packing is preferred.The zeroth-order theory also allows vesicles for v / a &


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606 M I C E L L E S , V E S I C L E S A N D M I C R O E M U L S I O N Sapproximation as equal to that for a (planar) bilayer. We have no reason to assumethat curvature is predominantly due to head-group interactions. There may well becurvature effects due to hydrocarbon tails of comparable magnitude which couldeven take a differentsign (vesicle-forming surfactants generally have relatively largehydrocarbon volumes).believe that vesicles are thermodynamically unstable (the stable state being thelamellar phase). Monodisperse vesicles of size given by the zeroth-order theory2 canbe produced through prolonged sonication. This is generally taken to indicate thatenergy is required for their production. When monodisperse vesicles are soproduced and cooled below the gel temperature, they become unstable and grow insize, eventually forming lamellar parti~1es.l~ n reheating, whatever size dis-tribution existed at the lower temperature is apparently frozen and seems to bekinetically stable.l3>l4Again it is claimed that with other preparation techniques anysize vesicle can be made to order. Not all these different distributions of vesiclescan be the thermodynamically stable state. All that can be inferred is that vesicledispersions are very slow to equilibrate, so that it is difficult to ascertain theequilibrium situation.On the other hand, on addition of excess water, the lamellar phase dispersesspontaneously to form multilamellar particles or liposomes6,17 ( e . g . lecithin) orvesicle^^"^^ (e .g . phosphatidyl serine). This would seem to suggest that the lamellarphase may not be stable in a dilute solution.Another problem which we have not yet addressed is that of interactions betweenbilayers. The question of stability of vesicles cannot be resolved theoreticallywithout proper account of interactions. For all these reasons the problem of vesiclesneeds to be reconsidered with care.

In addition the experimental situation is still unresolved. Some

4. V E S I C L E S4.1 P A C K I N G I N V E S I C L E S

We consider first consequences of packing conditions which apply under theassumptions of (1)constant volume 0 per surfactant molecule and (2) constant head-group area a. for a surfactant molecule. In general a vesicle (cf. ig.2) of aggregationnumber N is described by four parameters of the set V , a, , ai, I,, l i , Re, Ri, N , N ,, Ni,where the subscripts e, i refer to external and internal parameters. Of these we takethe first four v , a, = ai= ao, , to be given. We have

e

FIG.2.-Geometry of a model vesicle.

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D . J . M ITC H ELL A N D B . W . N IN H A M 609concave surface. We now compare the energies of cylinders and vesicles (with thesame u and ao). For cylinders we have

( 1 0 )2 R , 'where R, is radius of the cylinder. By contrast, for vesicles we have

where

Hence ps(vesicle) can be written

where the last form follows from elementary geometry. Comparison of eqn ( 1 0 )and( 1 3 )shows that cylinders will be energetically favoured over vesicles if and only if

We now establish conditions under which the inequality holds. To do this wewrite r e f R e / l e , r i 'Ri / le , x = RJRi SO that the inequality (14) becomesr : + r f 4u 1 + x2 4u>- or -) - .re -r i aOle 1- sol,

Now re, Ti and x are all functions of u / a & In fact from eqn (7)

and x is determined by eqn (9). Hence the condition eqn ( 1 5 )can also be written as

The solution of( 1- x ) ( l X 3 ) 3

(1+x2 )2 4- -then determines vesicle parameters for which the inequality is satisfied. Therequired root is x = 0.1896, corresponding to which value we have from eqn (9) andTherefore, vesicles which satisfy the conditions(1 6 ) v /ao le= 0.4883, v/aol i= 2.104, Re / le = 1.528, Ri l le = 0.289, l i / l . = = 0.232.

V U lisole a d i le->0.4883 or equivalently ---0.2321 (18)

will be energetically less favoured than the cylinder.

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610 M I C E LL E S , V ES I C L ES A N D M I C R O E M U L S I O NSWe now show that all vesicles satisfy this condition given the minimal packingcondition of section 4 . 1 , If u/aol, > 0.4883 the condition is satisfied since 1,s 1,. Thisis not a particularly interesting result since we know that cylinders are forbidden by

packing in any case for u/aol ,>0.5 . On the other hand if u/ ao l ,< 0 .4883 , theexternal packing condition 1, d , is not sufficient to discriminate, and a packingcondition on internal chains [cf. qn (18)] is required. Let us impose the requirementl i / lc> 0.2321. Then certainly l i / l c > 0 .2321 . Thus for u/aolc 0.2321 is sufficient to ensure that cylinders are again the favoured structure.Typically for a lipid of 1,= 20 A, the mean internal chain length of a vesicle which hasan energy equal to the allowed cylindrical structure would be as low as 4 .6 A. This isimprobable. The corresponding vesicle would be an absurd structure, with aggre-gation number N -y 26, Ni = 2 .To sum up: Within a framework which admits the zeroth-order theory as a firstapproximation, i.e. provided curvature is a perturbation, cylinders are always thefavoured structure for $< u/aol, 0.2321, appears to be eminently reasonable. Entropy which favoursthe smaller vesicles over cylinders has been ruled out as a major factor and curvaturealways dominates. (To quantify this conclusion roughly compare the entropiccontribution to pN of eqn (1)with curvature energies. Take XN= 1 as an extremeupper bound. Then even for a small vesicle N b500, Re = 50 A, y = 5 0 ,

For D as small as 1 A, the smallest possible characteristic length which has muchmeaning the curvature contribution to p ; would be [cf. eqn (13) ] >4Du/Re=20 x lO-kT.}5 . F U R T H E R C O N S I D E R A T I O N S O N V E SI CL ES -N O N - L I N E A R E X T E N S I ON S

5.1 V E S I C L E S us . B I L A Y E R SThe zeroth-order theory of section 4 .1 suggests that vesicles would be the

preferred structure in the range < v/aoZ,< 1 . However, we have shown that theweak packing criterion for hydrocarbon chains 1, < , is not sufficient. Recognition ofthe role of an internal packing constraint can dramatically alter the picture. We firstinvestigate vesicle parameters in the zeroth-order approximation with the constraintl i >a/,.The strong condition l i 3O.25 1, established in section 4 .2 provides onebound on a.Evidence from multilayer studies suggests that the upper bound on a,LY S 0.5, is a reasonable guess. Vesicle parameters for a = 0.5 are listed in table 2 .These results were obtained numerically by solution of eqn (6)-(9) with the addi-tional constraint Zi >d, .Note that fora = 0.5, as u /ao l ,decreases from unity (bilayer formation favoured)vesicle size diminishes to around v / a o l ,= 0.7. Thereafter the optimal vesicle sizeincreases until at u / a o l ,= CY = 0.5 bilayers are again the favoured structure. Theturning point can be read off from table 1 . Thus if

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D . J . M IT CH E LL A N D B . W . N I N H A M 6 1 1TABLE2.-EFFECT OF INTERNAL PACKING CONSTRAINTS ON VESICLES WHEN

a = a, , a = 0.5, u = 1063 A3, I, = 20 8,

(1 o>(0.97)(0.87)(0.79)(0.74)(0.69)(0.65)(0.59)(0.56)(0.54)(0.50)

5354.561687277829094.598106

a379314388665861799813700

00700108553628335476115a3

2020202020201814.31311.910

20 0018.9 12560015 4 20012.5 143010.8 76010 56010 58010 88010 130010 2 43010 00

00112 7002 4005602201301704207701720

00

Vesicle parameters with u = 1063A3 (corresponding to egg lecithin) and I, taken to b e20 A, with varying head-group area a,. a s taken to be 0.5.corresponding to rather stiff chains the turning point is v/aoZc= 0.8. Bilayerswould be formed for 0.5 < v/aoZc< 0.65 and revert to vesicles again for v/aoZ,>0.65. This remarkable phenomenon is purely a consequence of geometry and is notuniversal. Thus for very fluid chains, say a = 0.25, vesicle size decreases mono-tonically until v/aoZcb=.5, below which value cylinders are the favoured structure.

5.2 N O N - L I N E A R E X T E N S I O N SSubject to those complications just discussed which arise due to internal packingconstraints, both entropy and head-group curvature tend to favour vesicles ascompared with bilayers. We wish to explore how that conclusion holds within awider framework. That is, we now admit the possibility that curvature effects arelarge, so that the assumption a = a. of the zeroth-order theory would no longer betenable. A theory which covers this situation is necessarily model dependent andnon-linear, depending on unknown details of molecular interaction.Perhaps the simplest mathematical description of head-group interactions for

non-ionic surfactants is via the capacitor model: Imagine that D is the averagedistance between positive and negative charges which form a dipolar head group (i.e.$3 is the dipole moment where q represents unit charge). Then for a bilayer weshould have27rq2DpL = y a +- ea (19)

where E is the capacitor dielectric constant. (A similar model might be used todescribe ionic surfactants withD the Debye length.) Corresponding to eqn (1l ) ,wenow have for vesicles

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612where

M I C E L LE S , V E S IC L E S A N D M I C R O E M U L S I O N S

Recall that Re, R , and Ri are external, middle and internal radii, respectively, withNe and Ni the number of external and internal surfactant molecules. ObviouslyOptimhation of p t e s is a tedious but straightforward numerical problem. Typicalresults are given in table 3, which should be compared with table 2. (For definitenesswe take 4 = 4.8 x 10-l' e.s.u., E = 78, y = 50 dyn cm-' so that a0 = J(372D)A2,where D is measured in A.) The particular algorithm used was that developed byBrent.20

Re R , >Ri>D.

TABLE 3.-TYPICAL VESICLE PARAMETERS IN CAPACITANCE MODEL FOR HEAD-GROUP INTERACTION WITH a = 0.5, v = 1063A3, ,= 20 A

7.55 18 (0.97)10 (0.87)12 (0.79)14 (0.74)16 (0.69)18 (0.65)20 (0.62)22 (0.59)24 (0.56)26 (0.54)30 (0.50)

53 53 53 00 a3 20 2054.5 54.8 55.3 646 607 20 18.661 62 68 130 96 20 13.767 68 85 84 53 20 10.472 68.4 88.4 83 53 20 1077 67.6 89 86 55 20 1082 68.4 91 92.5 63 19.3 1086 72 94 112 84.6 17 1090 78 98 143 118 15 1094 84 100 200 173 13.6 1098 90 102 303 281 12.2 10106 106 106 00 00 10 10

a395 0003 43013001250136515703 1503 2705 80012 800a3

a383 000173043039044055095017803 7509 680a3

An internal packing condition l i 3 aZcis again necessary to obtain sensible results.Variation of a,, Ui from a0 is considerable at larger head-group areas. Similarly,predicted vesicle sizes can vary by up to a factor of 2 from the zeroth-order theory inextreme cases. Similar results emerge for different values of a. Despite thesequantitative differences it does appear that non-linearity does not affect the mainconclusions of the zeroth-order theory.We have to emphasize that whatever the nature of the forces between headgroups (and chains) which give rise to curvature, these may well inject non-lineareffects and precise quantitative predictions are not feasible at the present time.However, the general pattern which emerges from the zeroth-order theory seems tobe on fairly safe ground. Moreover while the theory strictly depends on fourparameters (v , ao, I,, a ) the substantial predictions rely only on the surfactantparameter v / u& and are independent of the detailed form of the forces. The vexedand delicate question of the stability of vesicles must remain open in general,depending as it does on chain stiffness (subsumed in the parameter a) nd thecompetition between curvature and interaction between bilayers. The answer willdiffer from surfactant to surfactant and depends on temperature.

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D . J . M I T CH E L L A N D B . W . N I N H A M 6136. M I C R O E M U L S I O N S

6.1 T H E R M O D Y N A M I C S OF M I C R O E M U L S I O N SWe now broaden our enquiry to include oil-water-surfactant systems. In theabsence of surfactant, oil and water will separate into two phases (unless the volumefraction of oil is extremely low or very close to unity). In this case the oil aggregate isarbitrarily large, its size being determined by the absolute amount of oil present.When surfactant is added, some surfactant molecules will go to the oil-waterinterface and lower the interfacial tension. With the addition of still more surfactant,several possibilities arise. Aggregates may appear in either the water phase or the oilphase. These aggregates may be micelles containing a little oil, dispersed in water orinverted micelles containing a little water, dispersed in oil; or large drops of oil (orwater) with surfactant at the interface, dispersed in water (or oi1).21,22 t is possible to

solubilize all the oil (or water). Another possibility is the occurrence of a three phasesystem: water, oil and so-called surfactant phase which contains all threecomponents.21 Once aggregates form, additional surfactant will be consumed in theformation of aggregates and will have little effect on the tension of the oil-waterinterface. We here use the term microemulsion to mean a thermodynamically stabledispersion of large oil and surfactant or water and surfactant aggregates. We makeno distinction between large swollen micelles and microemulsions. Microemulsionsare often mon odi~pe rse~nd the size of the drops is determined by the relativeproportions of oil, water and surfactant, not by the absolute amount of oil, in thesolution. These are by definition one phase systems.The thermodynamic approach developed in the previous sections can be exten-ded to deal with these systems. Microemulsion drops can be treated in a similar wayto pure surfactant aggregates and the distribution determined by the law of massaction. As for surfactants where a simpler picture is provided by the less rigorouspseudo-phase approximation; a simpler picture of microemulsions is also obtained ifwe ignore translational entropy of the aggregates.Recall that for a surfactant solution the aggregates which form (in the pseudo-phase approximation) are those with min p& and the concentration at which theyform (the c.m.c.) is given by

This result was derived as an approximation to the law of mass action, but it can beseen more directly as follows. If someof the molecules were arranged in non-optimalaggregates they could lower their total free energy by rearranging into optimalaggregates. If p


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014 M I C EL L E S , V ES I C L ES A N D M I C R O E M U L S I O NSIf G M , N = (M+N ) p ,N denotes the free energy? of an aggregate containing A4molecules of oil$ and N molecules of surfactant then clearly out of all possibleaggregates with the given composition X = N / ( M+ N ) hose which form must havethe min p G,N.The minimization is taken with respect to shape and size,etc., at fixedcomposition.The composition of aggregates will be uniform if g = min pk,N is a concavefunction of X ; .e. d2g/dx22 0. However, if g is not a concave function the solutionmay contain a variety of aggregates of differing composition11 (cf. ig. 3).

FIG. 3.-Diagram illustrating determ ination of aggreg ate compo sition: ( a ) g is everywhere a concavefunction of X. If chem ical potentia l of on e com ponent is g iven (e .g. po), omposition XA f aggregatesand chemical potential of other component is determined. (b) g is not everywhere concave. For aparticular value of po and p s the tangent touches the curve at two points. In this case a mixture ofaggregates of compositions XI,X2 ill occur. A syste m comprising aggregate s of in termed iate composi-tion would have a higher free energy gA than th e mixture gg .

If we wish to determine the composition of the aggregates given the overallcomposition of the solution it is necessary to determine the concentrations, orequivalently the chemical potentials, of the monomers. The general problem isdifficult. The easiest way to proceed isv ia an indirect route. We assume a value forthe chemical potential of one species. This allowsus to determine the correspondingvalue of the chemical potential of the other species. The composition of thecorresponding aggregate and overall composition of the corresponding solutionfollow. The procedure is iterated until the required overall composition is achieved.One special case, indeed one of much interest, can be handled with relative ease:Fora microemulsion solution in the presence of excess oil the chemical potential of onespecies, oil, is in fact prescribed.Suppose then that we are given the chemical potential po of the oil (i.e. of themonomers; and also of bulk oil when excess oil ispresent). We wish to show that theaggregates (and there may be more than one type) which can occur for this po arethose for which the expression ps(M,N )= (GM,NM p o ) / N is a minimum. The

t It is argued in the Append ix that (M+N ) F ~ , ~s the Helmholtz free energy + Po V,,, where Pois thej: Th e argum ent holds for water ins tead of oil.(1 Th e argume nt here paral le ls the argum ent that the Gibbs free energy of a mixture of two componentsdivided by the total num ber of molecules must be a concave function of the mole fraction X ; therwise adou ble tangent construction is required t o render the G ibbs free energy per molecule g concave.24 A ninterval for which d2g/ax2= 0 corresponds to phase separation.

(external) pressure of the solution.

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D . J . M I T C H E L L A N D B . W . N I N H A M 615corresponding value of the surfactant chemical potential ,US is equal to this minimumvalue; i .e.

The proof is as follows. If ps< nin p&4, N ) , hen G& , N >M po +N ps for all Mand N. That is, the free energy of every aggregate exceeds the free energy of themonomers obtained by dispersing the aggregate. But if ps> min ps(M,N ) , thenG& ,N


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616 M I C E L L E S , V E S I C L E S A N D M I C RO E M U LS I O N SThe variables a and R are specified through the relations

47r 3 4 v R &V = M v O i , + N v - R w , a =-3 Nso that the minimum ps(M,N ) can be obtained by minimizing with respect to R wand a in succession subject to the packing constraint 1s I,. The minimum a t fixeda isgiven b y the smallest value of R w allowed by packing. This corresponds to a dropwith 1 = I , and no interpenetration of oil and surfactant for v / a o l ,> 5 or a sphericalmicelle for v / a o l ,


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D . J . M I T C HE L L A N D B . W . N I N H A M 617TABLE 4.-(a) OPTIMAL PARAMETERS OF OIL-IN-WATER MICROEMULSIONS

ASSUMING a = ao, NO PENETRATION OF OIL INTO TAIL REGION

0.50.60.70.80.850.90.951

0.36 1.580.52 2.100.66 2.960.78 4.640.84 6.310.89 9.650.95 19.701 m

0.050.140.290.480.600 .720.861

Ro/lc= R w / l c- 1

TABLE 4.-(b) OPTIMAL PARAMETERS OF WATER-IN-OIL MICROEMULSIONS(a= ao,NO PENETRATION)

ulao lc x =R,IRw RWIL &,,,,, &., (inverted micelles)11.11.21 .31.41.51.61 .82.0

11.101.191.271.361.361.511.591.79

0010.325.313.642.802.801.951.711.26

10.760.600.480.400.400.290.250.57

0.400.340.290.260.220.200.180.140.12 Roll,= R w / l c -1

ignore head-group size. The volume fraction of water q5w and radius Rw of the waterspheres is easily shown to be q 5 w = R ~ / ( 3 u / a o + R w ) .et dma, be the maximumdistance of any point of the surfactant region from the centre of the nearest watersphere. Clearly d,,, s Rw+ , . For a given volume fraction &v, the minimum valueof d,,, obtains for a f.c.c. lattice and can be shown to be

Hence the largest volume fraction allowed is that corresponding to a f.c.c. lattice withdma, + Rw+ , . Eliminating Rw, the tabulated results follow.

6.3 E F FE C T O F C U R V A T U R E I N M I C R O E M U L S I O N F O R M A T I O N ( H . L . B . )If v , a. and I , can be estimated, table 4 gives some idea of the size of dropsexpected and therefore of design characteristics. The model is clearly deficient in atleast two important respects. One is its lack of oil specificity which problem we defer.The other is the effect of curvature which can affect allowed optimal structures insome circumstances. A mathematical description of curvature effects which spans

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618 M I C E LL E S , V E S I C LE S A N D M I C R O E M U L S I O N Sthe full gamut of expectations and trends at a qualitative level only can be formulatedas follows:

( 1 ) For non-ionic surfactants we have for the surface contribution to ps(M,N)

where, as for eqn (18 ) - (20 ) the bilayer optimal area can be characterized forconvenience in terms of a capacitor model for head group interactions, ao=d ( 2 n q 2 D / & y ) .As for our analysis of vesicles, for definiteness only we takea . = d ( 3 7 2 D / A 2 ) .( 2 ) An identical description can be used for ionic surfactants. Debye-Huckeltheory leads again to the capacitor model with D = K - ' , the Debye length, with

for a 1 : 1 electrolyte. no is the bulk electrolyte concentration. Taking E = 78for water, q = 4.8 x lo-'' e.s.u., k T = 4.116 x erg at 25 "C, we haveD = 3 .03c - l l 2 , where c is the salt concentration in mol dm-3. Taking furthery = 50 dyn cm-l as the surface tension of the hydrocarbon-water interface for roughestimates we have the correlation between salt concentration and head-group areaindicated in table 5 . The import of table 5 is that for ionic surfactants, saltconcentration or composition must provide a substantial contribution to head-groupcurvature and can markedly affect conditions under which microemulsions can form.

TABLE 5.-EFFECT OF SALT CONCENTRATION ON BILAYER HEAD-GROUP AREAc/mol dm-3 D / A a o / A 2 v / a o l c ( v / l c 30 A')

1 3.03 33 0.8930.5 4.29 40 0.7510.1 10.0 60 0.5020.05 13.6 7 1 0.4220.01 30.3 106 0.2820.001 96 189 0.159

Clearly both models of curvature encompassed in eqn ( 2 7 )are extreme and arenot to be taken at a quantitative level. However, characterization of curvaturethrough such models may be useful in two respects: ( a ) Assignment of D to aparticular non-ionic surfactant/water system can be used to predict behaviour forothers in a homologous series ( e . g . C,EO,); similarly for ionic surfactants withvarying salt concentrations; (b ) The very extremity of such models is useful if, givensuch flexibility, the substantial conclusions of the zeroth-order theory are confirmed.We consider then the form eqn ( 3 1 )and optimize p: with respect to a , takingR = 3 v / a for spherical micelles and Rw = R = Z c / [ $ - J ( 3 v / a l c - $ ) ]eqn ( 2 8 ) ] ormicroemulsion drops. Differentiating eqn (31 ) we have

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D . J . M I T CH E L L A N D B . W . N I N H A M 619where a aR

R aa--- - 1 spherica l micelles (R = 3 u / a ) ( 3 3 4

For v/aZ,> $ we have drops and fo r u/aZ,


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620 M I C E L L E S , V E S I C L E S A N D M I C R O E M U L S I O N SSeveral remarks are in order. Recall that if curvature is ignored, the zeroth-orderapproximation predicts (i) micelles v / a o l ,< 3 ; (ii) normal drops 3 < v / a & < 1 (iii)inverted drops v / a o l ,> 1. By comparison, when curvature is taken into account, the

situation is modified. Now we predict normal drops in a region where formerly wewould have expected inverted drops. For smaller values of v / a o l , curvature effectsgive larger drops, indeed can lead to drops where zeroth-order theory gives micelles.Note that for single-chained surfactants inverted drops only occur for low values ofa o , .e. very weak head-group interactions. The indications are that curvature plays alarger role in the determination of the dimensions of microemulsion drops than forvesicles, where there is a partial cancellation of curvature effects.It can easily be shown from eqn ( 3 2 ) - ( 3 3 ) that the condition for existence ofinverted drops is

and for micelles1 J ( l+2D/Z,)3 ( 1 + W C ) *v / a o l , 3, will have I = I,. However, the situation would not changesignificantly (except for vesicles) if 1 , represented an optimum length rather than amaximum length. It is essentially the tail length at which the tail free energy resistsfurther lengthening under the influence of head-group curvature energy.In the case of vesicles we have found it necessary to introduce a lower limital, onthe tail length. The free energy of the tails is assumed to rise rapidly if 1 falls belowal,. However, it would appear that tail lengths can vary by at least a factor of two.

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D . J . M I T C H E L L A N D B. W . N I N H A M 621The main omission in this lowest level theory is the possible effect of curvature onthe free energy of both head group and tail. We have explored this matter in sections4.2, 5 .2 and 6.2 .

7 .2 H . L . B . , A N D EFFECTS OF S A L T , T E M P E R A T U R E , A N DC O S U R F A C T A N T S

The packing ratio v/ao2,provides a measure of the hydrophilic-lipophilic balance(h.1.b.). For values of v / a o l , 1we predict inverse micelles and drops. That is, for v/aoZ,< 1 we predict atendency to curve spontaneously into normal structures and for v/ao2,> 1 to curvespontaneously into inverse structures.The packing ratio is affected by many factors including hydrophilicity of headgroup, ionic strength of solution, pH, temperature and the addition of lipophiliccompounds such as cosurfactants. The more hydrophilic the head group the strongerthe repulsion between the head group and the larger a . will be. The smaller the valuefor v/aoZc, the smaller will be the micelles formed. Thus (single chained) ionicsurfactants (at low salt concentrations) tend to form small spherical or globularmicelles whereas monoglycerides tend not to form micelles at all but form vesicles orliposomes or lamellar particles in dilute solution.12916The non-ionic surfactantspolyoxyethylene dodecyl ether (C12Em) form micelles at room temperature iftheir head groups are long enough ( e . g . C12E6, C12E8) ut C12E4 forms lamellardispersions. l 2The addition of NaCl to micellar solutions of sodium dodecylsulphate causes themicelles to grow in size and become long rod-shaped micelles.172g Similarly theaddition of HCl to micellar solutions of potassium oleate causes the micelles to grow,and, if the pH is lowered sufficiently, the solution goes cloudy and containslipo~omes.~n both cases the repulsion between head groups decreases, i.e. a .decreases and v / a o l , increases, so that our theory would predict a change fromspherical micelles to cylindrical micelles and eventually to vesicles.The incorporation of cosurfactants into surfactant aggregates would be expectedto increase the mean volume per surfactant molecule without affecting appreciablyeither a. or 2,. Consequently v/ao2,would increase with addition of cosurfactant,leading successively to an increase in micelle size, the formation of long rod-shapedmicelles and eventually the formation of lamellar phase or lamellar dispersion. 26,27The theory also predicts the observed increase in stabilization of hydrocarbons byionic surfactant solutions due to the addition of cosurfactants ( e . g . cf. Shinoda22).There is n o need to invoke specific interactions (which are discounted by Shinoda andFriberg,21 and by Wennerstrom and Lindman2628).The effect of temperature on v / a o l c is difficult to predict without a betterunderstanding of the forces at play. However, many of the observed phenomena forbiological lipids2*12*21s well as12,21 for non-ionic surfactants such as poly(oxy-ethylene) alkyl ethers may be accounted for if we assume v/aoZc increases withtemperature. Micellar size would be expected to increase with temperature. Non-ionic surfactants do tend to go cloudy when the temperature is raised above the aptlynamed cloudy point22 ndicating the formation of large aggregates [or flocculation ofaggregates, the phase diagrams of poly(oxyethy1ene) alkyl ethers display a lowerconsulate curve22 near the cloud point which indicates an attractive interactionbetween the aggregates above the critical point]. The surfactant C1ZE4 forms1 alamellar dispersion at still higher temperatures.

26

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622 M I C E LL E S , VE S I C LE S A N D M I C R O E M U L S I O NSIf v / a o l , increases with temperature one would expect the solubilisation ofhydrocarbons in non-ionic surfactants to increase with temperature (as observed)22until v / a o l c eached the value 1where phase inversion would be expected. At higher

temperatures water-in-oil microemulsions would be expected and the solubilisationof water would decrease as temperature rises (again as observed).21322As a. increases with increase in length of the poly(oxyethy1ene) head group wewould expect then an increase in the cloud point and phase-inversion temperature(p.i.t.) with head-group length (as observed).227 .3 O I L S P E C I F I C I TY

The model of microemulsion drops presented in section 6 ignored the possibilitythat the oil might penetrate into the surfactant interphase. Some phenomena of oilsolubilisation appear to require the assumption that hydrocarbons, some more thanothers, do penetrate into the interphase. As a consequence, the packing ratio v / a o l cis effectively increased, due to an increase in the interphase volume per surfactantmolecule.suggest that smaller alkanes penetrate into the surfactant layers more than largeralkanes. If we assume this to be true then this accounts for the variation withalkane chain length of cloud points of non-ionic surfactant solutions with saturatedhydrocarbon.22 This observation can be illustrated and quantified through thefollowing example. From the experiments discussed by Gruen and H a y d ~ n ~ ~e seethat decane swells the hydrocarbon layer of a glycerol monoleate black film to doubleits thickness (from 24 to 48 A). Experimentally it is known that the head-group arearemains constant. Thus the volume per surfactant molecule doubles and so u/aolcalso doubles. On the other hand, the p i t . (which corresponds to v / a o l c= 1)of thedecane-C12E4-water system31 is approximately 20 "C which is coincidentally theonset temperature of lamellar dispersion for aqueous solutions of CI2E4 2 cor-responding to v / a o l c = . z . Thus also in this example decane appears to swell thesurfactant hydrocarbon tail volume by a factor of two. With longer chain hydro-carbons v /ao lc changes less m a r k e d l ~ ~ ~ . ~ 'e . g . for hexadecane by a factor of $).Long-chain alkanes elevate the cloud point.22 This is to be expectea since v / a o l cis hardly altered, and our model predicts that large, non-spherical micelles will bereplaced by small microemulsion drops.

Experiments on the solubility of hydrocarbons in lipid lamellar

1

7.4 T H E I N T E R F A C I A L T E N S I O N C O N T R O V ER S Y A N DT H E S U R F A C T A N T P H A S E

Some have suggested that a microemulsion forms spontaneously as aresult of the interfacial tension between the oil and water phases becoming zero oreven taking negative values. If, on addition of surfactant or cosurfactant to thesystem, the interfacial tension becomes negative the system would tend to increasethe area of the oil-water interface, taking up surfactant from the solution, until theinterfacial tension returned to zero. The system would then be stable because: If thearea were to decrease, the interfacial tension would become negative, favouring anincrease in the area, whereas if the ;Irea were to increase the interfacial tension wouldbecome positive favouring a decrease in area. However, if we ignore such factors ascurvature energy, the system would be indifferent to whether an oil-in-water orwater-in-oil emulsion formed, or to what size distribution of drops formed, providedonly that the total area achieved the required value.

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D . J . M I T C HE L L A N D B . W . N I N H A M 623By contrast, if we invoke a curvature energy to determine the size of the dropsand to determine which phase is dispersed in the other, we find that the micro-emulsion will form while the tension of the oil-water interface is still slightly positive.This can be seen from the following argument:If curvature energy favours a dropof finite size it will do so only because this drop has the minimum value of

( G M , N -M,uo)/N or of (GM,NMpw) /N. The drops will form when the chemicalpotential of the surfactant is equal to this minimum value. On the other hand, for aninfinite drop of oil ( G M , N - M p o ) / N represents the interfacial Gibbs free energy persurfactant molecule and if the interfacial tension of the infinite drop is zero, thisinterfacial Gibbs free energy is also equal to the chemical potentialof the surfactant.Thus the finite drop will form at a surfactant chemical potential lower than thechemical potential at zero interfacial tension, i.e. when the interfacial tension ispositive.Shinoda and Friberg2' sketch the variation of interfacial tension between the oiland water phases with temperature for a system containing equal amountsof waterand oil r47.5 wt YO, nd 5 wt Oh of poly(oxyethy1ene)nonylphenylether]. The inter-facial tension drops to a very low value at the p i t . This might be understood roughlyfrom our model if we assume again that v/aol , ncreases with temperature, taking onthe value unity at the p i t . Below the p i t . v / a o l c< 1,curvature energy favours an oilwater dispersion and this implies a slightly positive interfacial tension. At the p.i.t.v / a o l c= 1, curvature energy favours zero curvature leading to a zero interfacialtension. At and near the p.i.t. therefore one expects either large drops with littlecurvature or an essentially lamellar structure, or some such bicontinuous phase.3773*Such large structures would be expected to interact strongly and form a separatephase; hence the surfactant phase which is commonly observed2' at the p i t . Theabove argument and the experimental data suggest then that not only can micro-emulsions form at positive interfacial tension, but that a positive tension is in factrequired. Very small interfacial tensions will occur only when the surfactant phaseforms.

7 .5 I N T E R A C T I O N SSo far we have not taken interactions between aggregates into account. Thetheory has been restricted to dilute oil-in-water dispersions in the presence of excessoil, or vice versa. We have correlated the type of microemulsion which forms to thetype of aggregate which forms in the absence of oil. If we ignore the penetrationof

oil into the surfactant tails we would predict that surfactants forming normal micellesor vesicles should, on addition of oil, form small swollen micelles or microemulsiondrops; and that surfactants which form inverse micelles should form water-in-oildispersions on the addition of oil. The type of aggregates formed by surfactantsolutions can change from normal to inverse structures on increase in temperature oraddition of cosurfactant. Penetration of oil into the surfactant tails will shift thebalance towards inverse structures.Changing the water content of a surfactant solution can also change the type ofstructure which is observed.12727 here are a number of possible related reasons forthis. Firstly, for a fixed volume fraction the distance between aggregates can beincreased by decreasing the area per head group which, through the packingconditions, can change the shape from sphere to cylinder or from cylinder tolamellar. Increasing the distance between aggregates may decrease the (repulsivehydration or ionic) interaction energy.33 Secondly, the distance between aggregatescan be increased by changing the shape from spherical to cylindrical or fromcylindrical to lamellar, thus decreasing the interaction energy at the expense of

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624 MICELLES, VESICLES AN D MICROEMULSIONScurvature energy. Thirdly, decrease in water content may reduce the amount ofwater available for hydration of the head groups. This would lead to a smallerhead-group area, larger value of v / a& and thereby could possibly induce a change inshape of the structures present. Whatever the reasons, change in structure isobserved in surfactant solutions with change in the concentration. One mightreasonably expect that analogously to the dilute solutions considered in this paper,the type of microemulsion formed on the addition of oil might be correlated with thestructures present in the absence of oil, taking into account the effects of penetrationof oil into the surfactant tails. Since surfactant solutions can change from normalmicellar solutions to lamellar phase to inverse micellar solutions, on decreasing thewater content, one might reasonably expect that water-oil-surfactant-cosurfactantsystems may change from oil-water microemulsions to water-oil microemulsionsvia a lamellar phase as is indeed commonly Some surfactant solutionsform reversed hexagonal phases2 so that it is not surprising to find reverse hexagonalphases in oil-water-surfactant-cosurfactant systems.34We are well aware of controversies and debate on opposing points of view,Although the older mixed-film theory has merit and embraces part of the story,Friberg and B~ r a cz en s k a~ ~ave demonstrated that the distinction between micro-emulsions and swollen micelles is artificial. O u r object in this paper has been todevelop a unified approach to the subject of microemulsions which also embracesmicellar systems. On that unremarkable note we rest our case.

We owe a debt to several people: The problem was posed to us by J. Th.G. Overbeek. Extensive discussions with G. J. T. Tiddy, our colleagues J. N. Israel-achvili, S. MarEelja and B. A. Pailthorpe are gratefully acknowledged. We arealso much indebted to S. Friberg and K. Shinoda not just for the care and thorough-ness of their experiments, but also because of the clarity of their expositions on thesubject. B.W.N. is grateful to S. Friberg for much encouragement.

A P P E N D I XT H E L AW O F M AS S A C T I O N A N D T H E C O N C E PT OF A N A G G R E G A T EAs already emphasized in section 2 , the whole subject of association colloids isbased on intuitive notions and is fraught with difficulties which devolve on

definitions. The treatment in this paper is largely centred around the law of massaction and relies heavily on the notion of an aggregate (micelle, vesicle, drop, etc .) .Here we wish to explore the difficulties and limitations involved in these concepts andthe nature of the approximations made.The concept of an aggregate is not precisely defined. There is a certain arbitrari-ness about any mathematically precise definition but any definition adopted mustconform closely with our intuitive concepts and the predictions of the theory mustnot depend critically on our (arbitrary) choice of definition.(1) A ONE-DIMENSIONAL ILLUSTRATION. The difficulties alluded to are perhapsmost clearly illustrated by means of a simple one-dimensional model which allowsanalytic solution. Consider a one-dimensional gas with nearest-neighbour inter-actions. Its is

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D . J . M I T CH E L L A N D B . W . N I N H A M 625where p is the chemical potential, P = l /kT, p is the pressure and u ( x ) is thepotential of interaction between nearest neighbours. [We assume that the distancex is normalized by some suitable length such as the diameter of molecules so thatthe integral in eqn (Al) is dimensionless. The pressure p is then also scaledappropriately.] Suppose that

I P Ru ( x ) = o forx>R and '1 exp[-pu(x)]dx>>l. (A2)R OThen for PpR


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626 MTCELLES, VESICLES A N D M I C R O E M U L S I O N Sone-component system, i.e. ignore any solvent. The thermodynamic potentiala= PV is related to the grand partition function by

where ZN V) is the canonical partition function which is defined by

where h = ( 2 ~ f i ~ / m k T ) ~ ' ~s the thermal wavelength, rl, r2, . . . , N are the coor-dinates of the N particles and the Hamiltonian is taken to beN

HN = P ; / 2 m + UN(rl ,2, . . . ,r") .i = lAssume now that some arbitrary criterion has been chosen for defining anaggregate. For example we might agree that m molecules form an aggregate ofaggregation number rn (an m-mer) provided that no molecule of the aggregate ismore than some prescribed distance R from some other molecule of the aggregateand no subgroup of molecules is further than R from the rest of the aggregate, andprovided of course that the rn molecules do not belong to a larger aggregate. Thenfor each configuration (r l, r2 , . . . , N ) the N molecules partition into aggregates. Letn , denote the number of m-mers for rn = 1 , 2 , 3 , . . .,N. The partition functionZN(V)can then be rewritten as a sum over all configurations consistent with thepartition, or

where %! ({n,}) denotes the region of configuration space consistent with the parti-tion into nl monomers, n2 dimers, . . . n, rn-mers, etc. We have included in eqn(A13) a combinatorial factor which gives the number of ways of partitioning Nindistinguishable molecules into n , aggregates of size m,m = 1,2, . . . ,N. Any twoconfigurations which differ only by the interchange of two molecules should beconsidered identical. The necessary approximations consist now in ignoring inter-actions between aggregates and extending the range of integration in eqn (A12)to allow aggregates to move through each other. The errors induced by theseapproximations is negligible, and allows a desirable simplification. Thus we have

or

where

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D . J . M I TC H EL L A N D B . W . N I N H A M 627the integration being over all configurations satisfying the constraints imposed by thedefinition of an aggregate. If we recall that the set nm are subject to the constraint1, mnm = N , we have from eqn (A10)

Thus

This is equivalent to the law of mass action provided we take

For the above result to have any validity it is necessary not only that the system besufficiently dilute that interactions between aggregates (and monomers) be negligiblebut also that I , be independent of the precise definition of an aggregate. Thisrequires that the potential of interaction be short-ranged and strongly attractive.The thermal wavelengthA does not play any fundamental role in the above argumentexcept (primarily) as a length scale to keep the above equations dimensionallycorrect. Any other suitable scaling length might be chosen, e.g. the diameter of amolecule ( u ) . f we rewrite eqn (A18)and (A19)

we will have merely replaced y by y +kT In ( A / U ) ~ and m y : by m y ; +( m- 1)kT In ( A / u ) ~ . That is we have merely changed concentration units andchanged the reference state with respect to which the chemical potential is defined.Thus the classical potential of the ideal gas is nowkT In ( p u 3 ) nstead of kT In (PA ').The reference state is the hypothetical ideal gas of densityp = l /u3 .This change isconvenient when working at a fixed temperature but is awkward when consideringchanges in temperature since the reference state is temperature dependent.(3) THREE-DIMENSIONAL, MULTICOMPONENT SYSTEMS. The above argument canbe extended to a multicomponent system. For example consider an aggregatingsolute B dissolved in a solvent A. We may write the grand partition function as

( A mwhere p = l/kT, P is the pressure, V the volume, pAand p B he chemical potentialsof solvent and solute, respectively, A A and A B the thermal wavelengths anduNA,NB(q, ) the potential energy of the NA solvent molecules and NB solute

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628 M I C E LL E S , VE S I C LE S A N D M I C R O E M U L S I ON Smolecules whose positions are given by q and r, respectively. If we letPodenote thepressure of pure solvent at the given pA and T hen

We define the excess free energy CTNB(r)f the NB solute molecules as

so that eqn (A22)may be rewritten as

where 7r = P -Po is the osmotic pressure of the solution relative to pure solvent atpressure P and temperature T. Eqn (A25)is completely analogous to eqn (A10)and(A11 ) with the gas pressure P eplaced by the osmotic pressure 7~ and the potentialenergy of the gas molecules replaced by the excess free energy of the solutemolecules. The same argument may therefore be applied to obtain the law of massaction for the solute and obtain formal expressions for thepk. Thep k are thus givenas functions of pA and T but for dilute solutionspA s approximately that of the puresolvent at the appropriate pressure (usually 1 atm).So far we have only obtained a formal expression for pk. In the next subsectionwe shall attempt to obta in a physical interpretation for pk .(4) PHYSICAL MEANING OF pk. We have obtained a formal expression [eqn (A19)]for p k in terms of the partition function I,,, defined by eqn (A16). The partitionfunction I , resembles a canonical partition function except that for our integral1, isover configurations which constitute an aggregate whereas the canonical partitionfunction is an integral over configurations in which the molecules are confined to afixed volume. However, as is well known in statistical mechanics, the majorcontribution to the canonical partition function comes from configurations with auniform density filling the whole container. One might likewise expect that themajor contribution to I , comes from configurations with a uniform density equal tothe average density of an aggregate and that I , would be approximately equal to thecanonical partition function of a system of m particles with this average density. Them p z would be approximately the Helmholtz free energyof a system of rn moleculesof this average density. Since the system is of finite size it will have in addition to abulk free energy term a surface energy and possibly curvature energy.It should be clear from the above example that the constraints defining anaggregate should confine the molecules to a sufficiently small volumeso that a naturalaggregate will form spontaneously just as a gas if confined to a small volume willcollapse to form a liquid drop.The interpretation of p k for a solution is more involved. The expression [eqn(Al6)] or I,,, will involve the excess free energy defined by eqn (A24). If we assumephase separation of the solute and the solvent then clearly rnpz equals the bulkHelmholtz free energy for ?n molecules at the average aggregate density+PoVagg+surface terms+curvature terms. If the aggregate is incompressible the sum of thefirst two terms is the bulk Gibbs free energy of m molecules at pressure Po [ i .e .at theexternal pressure (usually 1atm) not at the internal pressure of the aggregate which

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D . J . M ITC H ELL A N D B . W . N I N H A M 629may differ from Po by the Laplace pressure 2 y / r where y is the interfacial tension andr the radius of curvature of the drop].

J. N. Israelachvili, D. J. Mitchell and B. W. Ninham, J. Chem . SOC.,Faraday Trans. 2, 1976 ,72 ,1525.J. N. Israelachvili, D. J. Mitchell and B. W. Ninham, Biochim. Biophys. Acta , 1977 ,470 ,185 , seealso J. N. Israelachvili, S. MarEelja and R. G. Horn, Quart. Re v. Biophys., in press.C. Tanford, The Hydrophobic Effect (John Wiley & Sons, New York, 1973).D. G. Hall and B. A. Pethica, in Nonionic Surfactants,ed. M. J . Schick (Marcel Dekker, New York,1967).' G. S.Hartley, Trans. Faraday SOC.,1941, 37, 130.ti F. H. Stillinger, J. Solution Chem ., 1973, 2, 141.' . Y . C. Chan, D. J. Mitchell, B. W. Ninham and B. A. Pailthorpe, in Wa ter, a Comprehensioe' . Wulf, J. Phys. Chem., 1978 ,82 , 804.Treatise, ed. F. Franks (Plenum Press, New York, 1979), vol. 6 .

J. A. Reynolds, D. B. Gilbert and C. Tanford, Proc. Nut1 Ac ad . Sci. U S A , 1 9 7 4 , 7 1 , 2 9 2 5 .C. Tanford, J. Phys. Chem., 1972,7 6, 3020.G.N. Gershfeld, An n. Rev. Phys . Chem., 1976, 27, 349.G. J. T. Tiddy, Phys. Rep., 1 9 8 0 , 5 7 , 1.A. L. Larrabee, Biochemistry, 1979,18 , 3321.D. A. Gingell and L. Ginsberg, in Mem brane Fusion, ed. G. Poste and G. L. Nicholson (ElsevierNorth-Holland Biomedical Press, Amsterdam, 1978), p. 79 1.J. M. H. Kremer, M. W. J. van der Esker, C. Pathmamanoharan and P. H. Weisema, Biochemistry,1 9 7 7 , 1 6 , 3 9 3 2 .D. Papahadjopoulos, Biochim. Biophys. Acta, 1967, 135 ,624 .W .Helfrich, 2.Naturforsch., Teil A , 1978, 33, 305.J. S. Clunie, J. F. Goodman and P. C. Symons, Trans. Faraday SOC.,1969, 65, 287.tives (Prentice-Hall, New Jersey, 1973), chap. 7 .K . Shinoda and S. Friberg, Adv. Colloid Interface Sci., 1975, 4, 281.J. Th. G. Overbeek, Farad ay Discuss. Chern. SOC.,1978 ,65 , 7.

101 1121314

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l6 K. Larsson, 2. hys. Chem. (N.F.) 1967,5 6, 173.1718192o R. Brent, in Algorithms for Finding Zeros and Extrema of Functions without Calculating Deriva-2122 K. Shinoda, Principles of Solutio n and Solubility (Marcel Dekker, New York, 1978).24 G. S. Rushbrooke, Introduction to Statistical Mechanics (Oxford University Press, 1949).2 5 W. R. Hargreaves and D. W. Deamer, Biochemistry, 1978 ,17 , 3759 .23

H. Wennerstrom, J. Colloid Interfa ce Sci., 1979, 68, 589.P. Ekwall, in Advances in Liquid Crystals, ed. G. Brown (Academic Press, New York, 1975),vol. 1 , p . 1.H. Wennerstrom and B. Lindman, Phys. Rep., 1979 ,52 , 1 .D. W. R. Gruen, Biochim. Biophys. Ac ta, 1980,5 95, 161.S . Friberg, I. Buraczewska and J. C. Ravey, in Micellization, Solubilization and Microemulsions,ed. K. L. Mittal (Plenum Press, New York, 1976), vol. 2.Microem ulsions, Theory and Practice, ed. L. M . Prince (Academic Press, New York, 1977).

2627

2829 D. W. R. Gruen and D. A. Haydon, Pure App l. C hem., 1980, 52, 1229; Biophys. J. , in press.3031

3233 V. A. Parsegian, N. Fuller and R. P. Rand, Proc. Nut1 Ac ad . Sci. U S A , 1 9 7 9 , 7 6 , 2 7 5 0 .34 Microem ulsions, Theory and Practice, ed. L. M . Prince (Academic Press, New York, 1977), p. 91and references therein.Micellization, Solubilization and Microemulsions, ed. K. L. Mittal (Plenum Press, New York,1976), vol. 2, p. 797.36 D. Y. C. Chan, D. J. Mitchell, B. W. Ninham and B. A. Pailthorpe, J. Chem.SOC., araday Trans.2, 1978 ,74, 1669. See also E. H. Lieb and D. C. Mattis, Mathematical Physics in On e Dimension(Academic Press, New York, 1966).L. E. Scriven, Nature (London) ,1976, 263, 12 3.

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3738 J. N. Israelachvili and J. A. Wolfe, Protoplasma, 1980, 100, 315.

(PAPER 0 /984)

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