028_langmuir_1988_4_802

8/13/2019 028_langmuir_1988_4_802

1/5

802 Langmuir 1988,4, 802-806Theory of Microemulsions: Comparison with ExperimentalBehavior+

M. E. Cates*In st itu te for Theoretical Physics, U niver sity of California, Sa nt a Barbara, California 93106

D. Andelmans and S. A. SafranCorporate Research Science Laboratories , Exx on Research and Engineering C om pany ,

Annanda le , New Jersey 08801D. Roux*JDep artm ent of Chemistry and Bioche mistry, U niversity of California at Los Angeles ,Los Angeles, California 90024

Received J anu ary 26, 1988. I n Final Form: Marc h 24, 1988A thermodynam ic model for microemulsions (previously described) is used to predict some furth er aspectaof the p hase behavior. Th e stability of the m icroemulsion under variation of molecular par am eter s (suchas he bending constant and spontaneous curvature of the surfactant monolayer) is discussed. A n appropriatecut through the multidimensional parame ter space yields the characteristic fish phase diagram,as oundexperimen tally. Our model exhibits a phase beh avior close to th e exp erim enta l one, not only in term s ofthe v ariation with surfac tant concentration but also in terms of t he sen sitivity of t he phase diagram tothe n atur e of th e surfa ctan t, particularly its spontaneou s curvature.

1. IntroductionIn this paper, we study t he phase behavior of m icro-emulsions using a simple thermodynamic model. Th emodel has been described in de tail previously;' here wefocus on some aspects of ph ase stability tha t appear par-ticularly relevant to recent experimentalSpecifically, in section 3 we consider th e case when oil andwater volume fractions are equ al and study the evolutionof phase equilibria under va riation of surfac tant volumefraction (a8),emperature T),and the elastic bendingmodulus KO)nd spontaneous curvature (C,) of the su r-fac tan t film. Th is film is modeled as an incompressiblefluid monolayer, which c overs the (extensive) two-dimen-

sional interface between coherent domains of oil andwater.'In experimental terms, the elasticity param eters KO ndC, are expected to depend strongly on the particularsurfactant selected, on the type and concentration of co-surfactant, on salt concentration, and also on temp erature.While the relationships involved are ra ther complex, oneexpects qualitative comparison with experiment to bepossible with out knowing them in detail. For exam ple, ifthe salt and /or cosurfactan t concentration is chosen so asto o btain a m aximally symmetric pseudoternary phasediagram , we expect this to correspond to a case when th emonolayer has little or no spontaneo us curvatu re (Co= 01.lBy making small changes in salt an d cosurfactant con-centration, one can trace out some path in the Ko,Co)plane. To a first approximation, it seems reasonable toassume the salt concen tration affects mainly Corather thanKO, hereas the cosurfactan t concentration probably in-fluences both parame ters at once.g In this way, a corre-

'Presented at the symposium on Fundamental and Applied As-pects of Microemulsions, 11 , 61st Colloid and Surface ScienceSym-posium, Ann Arbor, MI, June 21-23, 1987; E. Kaler, Chairman.*Present address: Cavendish Laboratory, Madingley Road, Cam-bridge, CB3 OHE United Kingdom.IAlso at Lab. de Physique de la Matiere CondensBe, College deFrance, 75231 Paris Cedex 05, France. P resent address: PhysicsDepartment, Tel-Aviv University, Ramat Aviv, 69978 Israel.Present address: Centre de Recherche Paul Pascal, DomaineUniversitaire, 33405 Talence, France.

spondence can be made between th e parameters of themodel and experime ntal variables. Moreover, in somecases it is possible to s tud y experimen tally the elasticityparameters; '-12 thes e measure men ts, when available, ap-pear consistent with th e various trends predicted by ourmodel, as outlin ed in ref 1and f urthe r elucidated in section3. Expe rimentally, two properties are of special interest.A microemulsion system is specially useful when th e p hasediagram presents a continuous path from pure oil to purewater. T hat means that adaptin g the amo unt of surfactantor surfactant-cosurfactant mixture allows one to preparemicroemulsions of an y water to oil ratio. Th e efficiencyof th e microemulsion can be characterized by the minimumamou nt of surfactant (a8 ) eeded to get a stable (one-phase) m icroemulsion phase for a mixtu re of equal amountof water and oil. It is of practical im portanc e to make thisvalue asrns small as possible, since th e cost of an y com-mercial microemulsion is directly related to th e am ountof surfac tant needed to ma ke it. Generally for a givensystem, th e amount of surfac tant anrns optimized whenthe phase diagram is symmetrical in oil and water; th emicroemulsion is the n called balanced.In the case of terna ry systems with nonionic surfac tant,an elegant way of representing such a behavior has beenproposed by Kahlweit and colleagues.2d For these sy stems(1) Andelman, D.; Catea, M. E.; h u x , D.; Safran,S.A. J Chem. Phys.

1987,87,7229. Safran, S. A.; Roux, D.; Cabs, M. E.; Andelman, D. Phys.Rev. Let t . 1986,57, 491.(2) Kahlweit, M.; Strey, R.; Firman, P.; Haase, D. Langmuir 1985,1,281.(3) Kahlweit, M.; Strey, R. Angew. Chem., Int. Ed. Engl. 1985,24,654.(4) Kahlweit, M.; Strey, R. J. Phys. Chem. 1987, 91, 1553.(5) Kahlweit, M.; Strey, R.; Firman, P. J. Phys. Chem. 1986,90,671.(6) Smith, D. H. J. Colloid Interface Sci. 1984, 102, 435.(7) Smith, D. H. J. Colloid Interface Sci. 1985, 108, 471.(8) ang, J. In Physics of Amphiphiles, Micelles, Vesicles and Mi-croemulsionu; Degiorgio,V., Corti, M., Eds.; North Holland Amsterdam,1985.(9) de Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982,86, 2294.10) di Meglio, J. M.; Dvolaitaky, M.; Taupin, C. J.Phys. Chem. 1985,89, 871.(11) Meunier, J. J. Phys. Let t . 1985, 46, L 1005.(12) Safhya, C. R.; Row, D.; Smith, G. S.; Sinha, S K.; Dimon, P.;Clark, N. A,; Bellocq, A. M. Phys Rev. Let t . 1986,57,2718. Roux, D.;Safhya, C. R. J. hys Les Ulis, Fr.) to be published.0743-7463/88/2404-0802 01,50/0 1988 American Chemical Society

8/13/2019 028_langmuir_1988_4_802

2/5

Theory of Microemulsions Langmuir, Vol 4, No. 4 , 1988 803(linear in a,) has been adsorbed in to the surfactantchem ical potential. We define Coas positive for curvatu retoward water; note tha t Co = l /po where po is the spon-taneou s radius of curvatu re as used in ref 1.In eq 3, we have allowed for the f act th at t he effectivebending constant, K t ) , valuated a t the coarse-grainedlength scale f , may differ from the microscopic value, KO.This difference reflects the fac t th at the interfacial f ilmis subject to thermal fluctuations, corresponding to t heexcitation of undulation modes. These introdu ce a char-acteristic persistence leng th given by9f K = a e ( 4 ~ K ~ ) / ( a k ~ T ) ( 4 )where r is a num erical const ant of order unity. A t lengthscales comparable to or larger tha n th e persistence length,th e effective rigidity of th e surfactan t film is much re duce d,since the therm al undulations cause it to wander randomlya t this scale. An estim ate of th e effect is given by per-turb atio n theories,17 which p redict

(5)with T = akgr)/ 4.rrKo).n our calculations, we take r= 1 s the numerical constan t in eq 4 an d 5. In fact, th epredictions are quite sensitive to a ut this dependencecannot be separated from the variation with other nu-merical param eters of the model (such as the geometricalfactor 8n n eq 3, whose exac t values are equally uncertain ).Equat ions 1-3 and 5 complete th e specification of themicroemulsion bra nch of t he free energy surface, as afunction of the p arameters a, as, O, o, and T:

F p = Fbend(KO,CO,@r@s) T s ( @ , @ s ) (6)A detail.ed description of th e resulting p hase equilibria,as found by using a common tangent plane construction,is given in ref 1. Note th at, s ince salt and cosurfactantenter only through th e parameters KO nd Co, our resultsadm it a natura l pseudo ternary representation in terms ofthe oil, water, and s urfacta nt volume fractions.Obviously, the estimates given above for S and F b n d arenot very accu rate for extreme oil/water asymmetry (a i0 or i 1 . In these limiting cases, the system is moreaccurately described as a dilute solution of sur facta nt ineither w ater or oil, whose free energy may n ot be simplyrelated t o th e elasticity parameters, KO nd C of a nearlyflat monolayer. This is significant, because we are in ter-ested in two-phase (or three-phase) equilibria between amicroemulsion and a very dilute phase of surfactant inwater or oil (or two such phases). Qualitatively, however,as discussed in ref 1,we expect eq 1-3 to give a reason abledescription of the dilute phases, a t least in the case whenthe solubilities of the surfactant in water an d in oil are bothsmall and no t too different from one ano ther.To furth er characterize the phase equilibria, we may alsoconsider the possibility of ordered m esophases. Th e sim-

plest of these is th e lamellar phase, whose free energy maybe estimatedl as

K ( t ) = Ko[1- 7 n ( t / d I

OSFigure 1. Schematic representation of a fish. It correspondsto a cut of a ternaryphase diagram for aO/aw1. The one-phaseregion is indicated as 1; 2 and 3 correspond respectively to two-and three-phase equilibria.the temperature is the exper imental parameter whichcontrols th e balance of th e phase diagram. A cut of thephase diag ram is made a t a given oil/water ra tio (usually1) as a function of the temperature. Th e phase diagramhas a characteristic shape (named fish by th e au thors).Figure 1shows a schem atic representation of a fish. Th ebody corresponds to th e three-phase equilibrium, and th etail is th e one (microemulsion) phase. Th e characteristicshape indicates tha t asrns minimum for the symm etricmicroemulsion.

2. Model for MicroemulsionsTh e model presented here has evolved from earlie r workof Talm on an d Prager,13 Jouffroy, Levinson an d d e Gen-nes,14 and Widom.15 I t has been described in deta ilelsewhere; here we recall on ly the essential features.Th e microemulsion is characterized by a single struc tura llength scale, 5. For convenience, we describe th e stru cturein term s of a cubic lattice of side 5; the elemen tal cells ofth e lattice correspond to domains filled a t random w itheither water (probability a) or oil (probability 1 a). T hesurfactant resides in an incompressible monolayer at t heoil-water interface; taking t he thic kness of this monolayerto be a we find (using the random mixing approximation)

t / a = W1 - @))/as (1)Th is uniquely determ ines the cell size t in terms of th evolume fractions of surfa ctan t (as), ater aw 3 - @@/2),and oil (a,, = 1 CP - as). he en tropy density, accordingto this construction, is simply

S = -(1/t3)kB[@ In CP + 1 a) In 1- a)] (2)We now need to estimate the bending energy of themicroemulsion. As constructed above, th e curvatu re of th e

surfa ctan t film a t the oil-water interface is localized alongthe edges of th e elem entary cubes. A more rea listic esti-mate of the bending energy density may be found byround ing out these edges into sph erical sections, of radiusf / 2 . Doing this, we obtai n th e following form:Fbend = 8*/ t3)K t ) @ I - @)[I 2c0[ 1 2a ] (3)

This expression corresponds to a harm onic expansionabout the preferred curvature C0/2; an unimportant term(13) almon, Y.; rager, S J.Chern. Phys. 1978,69, 984.(14) ouffroy, J.; Levinson, P.; de Gennea, P. G. J hys. (Les Ulis,FT.)1982,43, 241.(15)Widom, B. . Chem. Phys. 1984,81, 030.(16)HeUrich,W. . uturforsch,A: Phys., Phys. Chem.,Kosrnophys.1973,28 93.

This represents a heurisitc generalization, to ourthree-co mpo nent syste m, of H elfrichs resultlg for th e free(17)Helfrich, W. J . Phys. Les Ulis, Fr.) 1985,46, 263; 987,48, 85.Peliti, L.; Leibler, S Phys. Rev. Let t . 1985,54,1690. ee also: Foerster,D. Phys. Let t . l986,114A, 15. Kleinert, H. hys . Let t . l986,114A, 63.(18) he result (5) s for Co= 0; here may naddition be a dependenceon Co which we neglect.(19)Helfrich, W. Z. aturforsch. A: Phys. Phys. Chem.,Kosmophys .1978, 3, 305.

8/13/2019 028_langmuir_1988_4_802

3/5

8/13/2019 028_langmuir_1988_4_802

4/5

Theory of Microemulsions

20 -ciM

IO -

Lan gm uir, Vol. 4 , No. 4, 1988 805

3 a ar1o . IO I O 0

@ SFigure 4. Experim ental data from ref 5 for a series of nonionicsurfactants (CiE.). his experim ental phase diagram has to becompared with Fwe 3. The s u m + is assumed to be directlyrelated to the elastic constant K . The open points correspond tothe l imit of stabili ty of the microemulsion phase toward thethree-phase equilibrium and the filled points to th e phase sep-aration with the lamellar phase.phase occurs a t smaller4 Since the slope of t he boundarybetween the m icroemulsion and the region of three-pha seequilibrium is larger than the slope of th at between themicroemulsion and lamellae, these two lines meet (for K= 8.3,asrn 4 X Above this value the re is no longera stable microemulsion phase, and the P region corre-spon ds to complex polyphasic equilibria between water,oil, lamellar phase, and microemulsion phases with awater/oil ra tio different from 1. Th is region correspondsin the ternary representation (at constant K ) to a loss ofthe co ntinuow path betw een oil and water. Consequently,the price to pay for having a micromem ulsion phase ma dewith a very small amount of surfactant is to have a narrowregion of stability. Th is illustrates the com petition be-tween small values of K , which favor microemulsion ph asesof small characteristic sizes (and which need a lot of sur-factant), and large v alues of K , which favor large structure s(microemulsionwith less surfactan t, bu t also the lamellarphase).Following these remarks, the recipe for making amicroemulsion that is stable over a wide range of surfactantconcentration is to make K small, in order to push thelamellar phase to higher as.However, asm ill then bequit e large, an d th e microemulsion will use a lot of sur-factant. On the o ther hand, if one wants a very small valueof K should be made as arge as possible while keepinga con necting single-phase pathway between the water a ndoil sides of th e apex of th e three-phase triangle, so t h a tth e pathway will be very narrow. Consequ ently, ther eexists a ra nge of values of K - - ~ ~ T )hich favors micro-emulsion phases (for Co = 0). Th is behavior is well il-lustrated by th e series of phase d iagrams of th e nonionicsurfactants given in F igure 3 of ref 5. Indee d, for smallsurfactant length, asrns large but there is no lamellarphase even for high concentration of surfactan t. On thecontrary, for large s urfactan t length (corresponding to ahigher elastic constant K ) , asrns much smaller and thelamellar phase comes much closer to th e three-phase tr i-angle. Th is is shown in Figure 4 where we took the dat afrom ref 5 and plot ted asrn,nd th e boundary with thelamellar phase, as a function of the su rfac tant length.22We have p lotted t he v alue of asrnor different surfactantcharacterized by the sum + (related to the total lengthof the surfactant). Th e length of the surfa ctan t is assume d

(22) The nonionic surfactants considered are CiEj,here is the car-bon number of the aliphatic chain and j the number of oxyethylenegroups. We used + to characterize the surfactant length.

I I

-15 I I I- 0 10 20 30@S

Figure 5. Cut of the phase diagram for a fixed value of the +.,/awratio (@o/@w = 1 . The value of K is kept constant ( K = &.6), ndCo ( l / po ) varies.to be directly related to the elastic constant K. The s im-ilarity between Figure 4 experimental data) a nd Figure3 (theory) is striking and suggests that our model describescorrectly the competition between the lamellar and mi-croemulsion phases.Th e second plot (Figure 5) is one in which the reducedspontaneous curvature Co (eq 10) is varied, along withsurfactan t volume fraction, again at a fixed ratio @,/Po= 1. Th e parameter K is held fixed. In th is case we finda l arge f i sh , a s seen e~ pe r im en ta l ly . ~ -~he head of thisfish is squashed against th e left-h and edge of t he figure.This fact is a result of th e approximation, made in themodel, th at t he dilute phases of surfa ctant in water an dsurfacta nt in oil can be represented by points on edges ofthe triangle of th e ternary representation. This could easilybe corrected by taking a more realistic description of th edilute phases. l Th e comparison with experiment, in therepresentation of Figure 5, is qualitatively easy to make,assuming that, in nonionic systems, the spontaneouscurvature is a smoothly varying function of tempe rature .A possible explanation could be th at t he w ater contentsin th e hydrophilic region of t he film decreased with tem -perature (the polar heads are less solvated at high tem -perature th an a t low temperature).

4. ConclusionsIn this pa per we have focused on compa ring more closelywith exp eriment the results of a previously proposed modelfor microomulsions. T he simp lest comparisons are withtrue ternary systemsof oil water, and nonionic surfactants;these are extensively studie d in the literature?+ We have

shown th at in term s of two phenomenologicalparametersK and Co we can qualitatively explain the d ependen ce ofthe phase behavior not only on the surfactant concentra-tion but also on its m icroscopic properti es, Le., sha pe an dlength. Indeed, a balance and optimized microemulsioncan be obtained from our model for Co = 0 and K in anintermed iate range, around a few kBT. Recently the modelhas been extende d to calculate the structure factor; thiscompa res well with X-ray a nd n eutron small-angle scat-tering ex perimen ts and with freeze fracture pictures.23 Fora more quan titative comparison with exp erimen ts, a m odel(23) Milner, S T.; Safran, S A,; Andelman, D.; Cates, M. E.; Roux,

D. .Phys. (Les Ulis ,Fr.) to be pu blished.

8/13/2019 028_langmuir_1988_4_802

5/5