One Hundred Years of Micelles

Transcription

1

2 One Hundred Years f Micelles Evlutin f the Thery f Micellizatin Ramanathan Naarajan Cntents. Intrductin...4. Early Qualitative Structural Mdels f Micelles First Quantitative Thery f Micelle Due t Debye Mnmer Micelle Equilibrium and Critical Micelle Cncentratin Wrk f Frmatin f Micelle Equilibrium Micelle Size Variance f Equilibrium Micelle Evlutin f Theries Addressin the Shrtcmins f the Debye Mdel Oshika Mdel Reich Mdel Emerence f A Thery fr Rdlike Micelles Halsey Mdel Statistical Mechanical Theries f Micelles Heve and Bensn Thery Pland and Scheraa Thery Micelle Shape Transitins and Size Distributin Cperative and Anticperative Free Enery Functin Representatin f Cperativity Cperativity and Frmatin f Cylindrical Micelles Alternate Representatins f Cylindrical Micelles Frmatin Tanfrd s Principle f Oppsin Frces Mlecular Packin Mdel fr Self-Assembly Nninic Surfactants Inic and Zwitterinic Surfactants Slutin Cnditins Duble Tail Surfactants Plar Oranic Slvents Mlecular Scale Predictive Theries f Micelle Frmatin Cntributins t Free Enery Chane n Areatin Transfer f the Surfactant Tail Defrmatin f the Surfactant Tail Frmatin f Areate Cre Water Interface Head Grup Interactins Steric

3 4 Surfactant Science and Technly..6 Head Grup Interactins Diplar Head Grup Interactins Inic Head Grup Interactins fr Olimeric Head Grups Estimatin f Mlecular Cnstants Needed fr Predictive Cmputatin Influence f Free Enery Cntributins n Areatin Behavir Mdified Packin Parameter Mdel Accuntin fr Chain Lenth Effects Extendin the Thery f Micellizatin Prspects...46 Acknwledments...48 References INTRODUCTION Surfactant mlecules are cmpsed f a plar head rup that likes water and a nnplar tail rup that dislikes water, thus cntributin t an intrinsic duality in their mlecular characteristics. Despite their mutual antipathy, the head and tail rups f the surfactant cannt leave ne anther because they are cvalently cnnected. The dilemma faced by these mlecules is reslved in nature by the intriuin phenmenn f mlecular self-assembly, wherein the amphiphiles self-assemble int three-dimensinal structures with distinct and separate reins cmpsed f the nnplar parts and the plar parts, havin minimal cntact with ne anther. Numerus variatins are pssible in the types f the head rup and tail rup f surfactants. Fr example, the head rup can be aninic, catinic, zwitterinic, r nninic. It can be small and cmpact in size r it culd be an limeric chain. The tail rup can be a hydrcarbn, flurcarbn, r a silxane. It can cntain straiht chains, branched r rin structures, multiple chains, etc. Surfactant mlecules with tw head rups (Bla surfactants) are als available and there are dimeric surfactants with tw head rups and tw tail rups with a cvalent linkae cnnectin thse (Gemini surfactants). Furthermre, the head and the tail rups can be plymeric in character, as in the case f blck cplymers. This variety in the mlecular structure f surfactants allws fr extensive variatin in their slutin and interfacial prperties and their practical applicatins. Therefre, it is nt surprisin that the drive t discver the link between the mlecular structure f the surfactant and its physicchemical behavir has had a ln histry. The existence f surfactant mlecules in the frm f self-assembled areates was first suested by McBain in 93 [] based n his studies n hw the cnductivity f a slutin f sap mlecules chanes with the cncentratin. Sap slutins exhibit even lwer smtic activity than wuld be predicted if ne assumed that sap existed in slutin as simple undissciated mlecules. Sap slutins als cnduct the electric current far better than wuld be expected frm the bserved smtic effects. Attemptin t explain these anmalies, McBain, in 93, suested that the fatty sap ins areated in slutin. Such cllidal areatins f ins, which were termed micelles, wuld explain bth the lw smtic activity and relatively hih cnductivity f sap slutins. Understandin the areatin prperties f surfactant mlecules has nearly a hundred years f histry. The early wrk, until abut 950, was sinificantly fcused n experimental methds t identify the size and shape f the areates. McBain [ 5], Adam [6], Harkins [7 ], Hartley [ 5], and Philippff [6,7] are amn the pineers wh prpsed different structural mdels fr micelles. The x-ray diffractin technique was extensively applied and there were many qualitative structural mdels prpsed fr the areates. The first thery, published in 949 by Debye [8 0], explained why micelles frm and why there are finite-shaped micelles. In the perid between 950 and 956, this wrk stimulated a number f theretical studies by Hbbs [], Oshika [], Reich [3], and Halsey [4], wh attempted t rectify sme f the fundamental theretical aspects as t hw an equilibrium system shuld be described and als drew attentin t the rle f surface enery, which was missin in the Debye mdel. Frm 955 t 965, further advances in thery emered and were uided by the methds f

4 One Hundred Years f Micelles 5 statistical mechanics. Specifically, the theries f Heve and Bensn [5] and Pland and Scheraa [6,7] have prvided sme key cncepts that have eventually been interated int the current mdels. Als durin this time, a liht-scatterin technique was applied t surfactant slutins and experimental evidence fr the frmatin f rdlike micelles was enerated. The wrk f Pland and Scheraa explicitly intrduced the rle f hydrphbic interactins in micelle frmatin. This cncept was quantitatively included in the free enery mdel prpsed by Tanfrd [8 30] durin 970 t 974. Tanfrd s mdel had the simplicity f the earlier thery f Debye, and was als cnsistent with a rirus statistical thermdynamic frmulatin f the areatin prcess ccurrin in the surfactant slutin. The free enery mdel incrprated all imprtant physical chemical factrs and prvided an explanatin fr why micelles frm, why they rw, and why they remain finite in terms f clearly identified ppsin interactinal effrts. Israelachvili et al. [3] used the framewrk f the Tanfrd free enery mdel aln with mlecular packin cnsideratins inside surfactant areates t develp a emetry-based apprach t predict the frmatin f different shapes f areates. They were able t explain the frmatin f spherical and cylindrical micelles and spherical bilayer vesicles, as well as transitins between these structures based n a cmbinatin f eneral thermdynamic principles, Tanfrd s free enery mdel, and the emetric cnstraints impsed by mlecular packin cnsideratins. At this stae, ne culd arue that the eneral principles f surfactant self-assembly were sufficiently well-established due t the cntributins f Tanfrd and Israelachvili et al., in particular (Fiure.). In 976, Kash Mittal ranized a sympsium n Micellizatin, Slubilizatin and Micremulsins [3 37], which became the precursr t the current biennial sympsia Surfactants in Slutin. These series f sympsia have iven impetus t practically all f the subsequent research in surfactant science and technly. In additin t stimulatin theretical wrk, the Surfactants in Slutin sympsia, since 976, have als cntributed t the extensive develpment f nvel applicatins usin surfactants beynd the classic areas f deterency, emulsins, fams, and dispersins. Chief amn these is the use f surfactants fr material synthesis, especially the use f reverse micelles as nanreactrs fr nanparticle synthesis includin metal xides, metals, and quantum dts and the use f surfactant liquid crystals as templates fr the synthesis f mesprus materials that vastly extended the pre size beynd the small rane pssible with zelites. The perid fllwin Tanfrd s wrk has seen a number f key theretical advances. First, theretical mdels bean t be develped with a view t a priri predict the self-assembly prperties frm the mlecular structure f the surfactant. Secnd, theretical mdels bean t be applied t mixtures f surfactants and als t surfactants that had hydrphbicity arisin frm FIGURE. Left, James W. McBain. (Curtesy f Stanfrd University.) Middle, Peter Debye. (Curtesy f Crnell University Archives.) Riht, Charles Tanfrd. (Cpyriht 009, The Prtein Sciety.)

5 6 Surfactant Science and Technly flurcarbns and silxanes. Third, self-assembly phenmena such as slubilizatin and micremulsins were described usin theretical appraches used fr surfactant areatin s that a unified view f self-assembly in multicmpnent systems was made pssible. Furth, sme key features f the free enery mdels, includin the calculatin f electrstatic interactins in inic surfactants, and treatment f hw hydrphbic chains pack and arrane inside areates f different shapes, were mre fundamentally treated. Fifth, the thery f self-assembly was quantitatively applied t nvel surfactants such as Bla surfactants and Gemini surfactants. Sixth, self-assembly phenmena at as liquid and slid liquid interfaces bean t be mdeled. Finally, the free enery mdels f the analytical frm were imprved and smetimes cmbined with free enery mdels develped by mlecular dynamic simulatins s that the predictive mdels can be truly a priri. In this chapter, we fcus mainly n the theretical evlutin f free enery mdels fr surfactant self-assembly. The mdels chsen fr discussin are imprtant in the sense that they have affected the evlutin f thery and have als influenced ur wn wrk in develpin predictive mdels fr a rane f self-assembly phenmena. Because f the structure f this chapter, a number f scientists wh have cntributed t experimentally identifyin imprtant phenmena r theretically mdelin ne r anther specific feature f the self-assembly will unmentined. Their imprtant cntributins and influence ver the evlutin f thery are nt t be inred and hpefully are visible thruh the extensive citatins f their wrk in ur previus theretical articles.. early QUALITATIVE structural MODELS OF MICELLES The early discussins n areate shapes (Fiure.) fcused n tw types prpsed by McBain [4,5], a lamellar (disk) areate with 50 t 00 mlecules and a small limeric surfactant cluster with apprximately 0 mlecules. The lamellar micelle cnsists f tw layers f sap mlecules r in pairs partially dissciated and arraned side by side, with the tw hydrcarbn layers inside. Harkins and cwrkers [7 0] interpreted their x-ray results as cnfirmin the existence f the lamellar McBain micelle. Furthermre, Hess and cwrkers prpsed the existence f McBain lamellar micelles that repeat in parallel arranement, separated by layers f water thus ivin an x-ray ln spacin equal t twice the lenth f the mlecule plus that f the layer f water [38,39]. This micelle was referred t as the Hess micelle (Fiure.). Harkins cnsidered hw mlecular shape influences mlecular packin at interfaces [6]. He initially favred the lamellar areate mdel and als prpsed a cylindrical lamellar frm in which Water Oil Water Water Water FIGURE. Left, Hess lamellar micelle made f tw McBain lamellar micelles; middle, Harkins cylindrical micelle mdel shwin his idealized crss-sectin f a sap micelle withut slubilized il and with slubilized il; riht, Hartley mdel f a spherical micelle.

6 One Hundred Years f Micelles 7 the surfactant mlecules are riented parallel t the axis f the cylinder, with the head rups cnstitutin the tw end surfaces f the cylinder (the cylinder bein merely a small sectin f the lamellar areate) [7 9]. Harkins als suested that althuh a cylinder seems t represent mst f the prperties f the micelle, it is imperfect in that it represents t lare an interface at the side between hydrcarbn rups and water. Thus, it des nt seem imprbable that a mdel, which lies between a cylinder and a cylindrical type f spherid, may be fund t mre accurately represent the enery relatinships. This wuld amunt t a distrtin f the side f the cylinder with which plar rups cver (t sme extent) the nnplar hydrcarbn chains [0]. Philippff [6,7] analyzed smtic activity, specific cnductivity, and x-ray data btained at varius electrlyte and surfactant cncentratins and fr varius surfactant tail lenths and identified tw reimes with respect t micellar shapes. In the first reime, which crrespnds t zer r small amunts f electrlytes, micelles are practically spherical in the sense that they need nt be true spheres but can be cubes, shrt cylinders, prisms, r spherids. In the ther reime, which crrespnds t lare electrlyte cncentratins and lner tail lenths f surfactants, the areates are anismetric and lare. Philippff cnsidered that it is imprbable fr small micelles t reranize int lare micelles with increasin salt and surfactant cncentratins and therefre pstulated the lare structures t be secndary areates f the prefrmed primary micelles. As ppsed t the lamellar structures, Adam [6] arrived at spherical micellar structure based n hw mlecules pack at interfaces dependin n their mlecular shapes. Adam suested that mlecules larer at their plar ends will naturally pack int a curved film havin the hydrcarbn side cncave and the water-attractin side cnvex. Hartley [ 5] prpsed the existence f larer spherical micelles with apprximately 50 surfactant mlecules. Hartley suested that the areates are essentially liquid and since they will tend t present the minimum surface t the water, they will presumably be ruhly spherical and f the larest radius cnsistent with nne f the heads bein submered in the paraffin interir. Harkins, wh initially supprted the lamellar micelle mdel and als prpsed a cylindrical lamellar structure, eventually cnsidered the pssibility f spherical shape fr micelles []. Crrin demnstrated [40] frm an analysis f the x-ray diffractin data frm the Harkins labratry that a lamellar mdel is nt required t explain the x-ray diffractin patterns btained frm slutins f ln-chain inic surfactants. Qualitatively, the diffractin patterns culd be satisfactrily interpreted by a mdel f spherical micelles whse relative psitin can be represented by a radial distributin functin. He arued that althuh this des nt prve the validity f the spherical micelle mdel, it indicates that x-ray measurements alne d nt allw ne t decide between the lamellar and spherical mdels. Durin that stae f develpment in surfactant research, n definitive thery f micelle frmatin yet existed. The fllwin statement frm Philippff [7] in his 95 article makes the case. Havin reviewed the field, we can make nly the neative statement that there is at present n thery t accunt fr the causes f micelle frmatin which can interpret the whle f the experimental evidence. Micelle frmatin is independent f the sin f the chare f the micelle frmin in. Micelles are partially inized, therefre a thery must accunt fr this. Micelles frm with nninizin deterents in water, and with sme inizin deterents in hydrcarbns (aersl OT, diethyl hexyl sdium sulfsuccinate), shwin that a chare is nt essential fr the phenmenn. A straihtchain cmpund is nt necessary fr micelle frmatin. Sdium dexychlate with a sinle inizin rup n a sterl skeletn frms micelles as well as divalent aersl OT in hydrcarbns with fur branched chains t ne inic rup. The cmmercially imprtant micellar arylalkyl sulfnates have a cmplicated structure f the hydrcarbn part f the mlecule. Likewise, an extended hydrphilic rup as in the plyethylene xide derivatives is nt prhibitive even t x-ray structures. Tween 40, plyethylene xide srbitan mnpalmitate, with even three hydrphilic chains, frms micelles. Mixed micelles are als readily frmed by deterents f the same eneral structure, differin nly in chain. The nly cmmn principle left is the sereatin f the hydrphilic and hydrphbic parts f a mlecule. It is in this cntext, that theries bean t be develped t explain ne r mre features f surfactant areates.

7 8 Surfactant Science and Technly.3 FIRST QUANTITATIVE THEORY OF MICELLE DUE t DEBYE The first quantitative, mlecular thery f the frmatin f micelles was prpsed by Debye [8 0]. He cnsidered inic surfactants cnsistin f a hydrcarbn chain with a chare at ne end. T create a micelle with the lamellar frm prpsed by McBain, Debye prpsed accuntin fr tw different kinds f enery. There is a ain in enery because a number f hydrcarbn tails are remved frm the surrundin water and bruht int cntact with each ther in the micelle. He cnsidered the mlecular frces f imprtance in this prcess t be relatively shrt-rane frces f the van der Waals kind. T brin the chared ends f the mnmers nearer t each ther n bth flat surfaces f the lamellar micelle requires enery t vercme the ln-rane electrical frces. Accrdinly, Debye cncluded that the interplay between shrt-rane van der Waals frces and ln-rane electrstatic frces are respnsible fr the equilibrium structure f the micelle..3. Mnmer Micelle Equilibrium and Critical Micelle Cncentratin Debye started with the classic mass actin equilibrium between simple ins and micelles. Cnsider the fllwin reversible assciatin equilibrium between fatty sap ins A (mnmers) and micelles A, where is the number f fatty ins present in a micelle and K is the equilibrium cnstant fr the mnmer micelle assciatin: K A A (.) If X dentes the cncentratin f micelles, X the cncentratin f unareated paraffin chains, and X T the ttal cncentratin f fatty ins, with all cncentratins expressed in mle fractins, then X T = X + X (.) Dentin the micellizatin equilibrium cnstant K in terms f a cncentratin X C in the frm, K = X, the applicatin f the mass actin law assumin unit activity cefficients leads t C X = K ( X ) = X C X X C X X X, T = + C X. (.3) X C Frm this mass actin law, fr lare enuh, ne can see that when X < X C, the micelle cncentratin X will remain neliibly small and X T will be practically equal t X. T prevent the diverence f the riht-hand side f the equatin, it is clear that X will becme equal t X C but never exceed it. Therefre, when micelles frm, X will be practically equal t X C. These cnsideratins indicate that if is lare enuh, the sap is practically unareated up t the cncentratin X C and, abve that cncentratin, all excess sap will appear in the frm f micelles. On this basis, ne can identify X C as the critical micelle cncentratin (cmc)..3. Wrk f Frmatin f Micelle T estimate the enery chane n micelle frmatin, Debye cnsidered the wrk necessary t create an areate frm sinle mlecules. One part f this wrk is electrical and it was estimated as fllws. The ttal chare n the micelle surface will be prprtinal t σ e R, where R is the radius f the micelle surface viewed as a circular disk and σ e is the cnstant surface chare density. The ptential at the rim f the disk will be prprtinal t σ e R. Therefre, fr a differential chane in the disk radius R, the chane in surface chare is πrdrσ e and additinal wrk shuld be dne aainst the Culmb

8 One Hundred Years f Micelles 9 frces prprtinal t R drσ e. The ttal electrical wrk invlved in buildin up the disk is thus prprtinal t R 3. Because the surface area R is prprtinal t the number f mlecules cnstitutin the micelle, the electrical enery t be vercme in creatin a micelle f size is W e = 3/ w e (.4) where w e is a fundamental electrical enery (absrbin all ther cnstants independent f the size ). Due t the ln-rane character f the electrical frces, this wrk increases faster than the number f mlecules in the micelle. The secnd wrk is related t the enery ained by brinin hydrcarbn tails in cntact (which invlves nly shrt-rane mlecular frces) with ne anther. It was represented as W m = w m (.5) with the intrductin f anther fundamental mlecular enery w m. The neative sin indicates that this enery represents a favrable prcess f brinin the surfactant s tail frm cntact with water t cntact with ther tails..3.3 equilibrium Micelle The ttal enery W = W e + W m, has a minimum fr a certain value = and at this pint, the enery W f the micelle is neative. This means that the micelle is mre stable than separate mlecules and that wrk is required t either increase r decrease the equilibrium number. The enery cntributins W e and W m and the ttal enery W are shwn in Fiure.3 as functins f micelle areatin number. The ttal enery W is neative with a shallw minimum W at. Frm the minimizatin f the ttal enery W, ne ets the equilibrium areatin number and the enery W crrespndin t this equilibrium areate. wm = W we wm wm 3/ w, = =, (.6) 3 e Enery per micelle (in kt) W e electrical W m van der Waals W ttal enery Areatin number FIGURE.3 Debye mdel enery cntributins t micelle frmatin. The electrstatic enery increases faster than the increase in the van der Waals enery as increases and thereby cntributes t the finite size f micelles.

9 0 Surfactant Science and Technly Equatin the enery f frmatin f the equilibrium micelle t the equilibrium cnstant fr the areatin in the mass actin mdel, and recnizin that the number f mlecules in the equilibrium areate will be much larer than, ne ets an expressin fr the cmc. kt ln X W = kt ln K = ( ) kt ln X kt ln X C C C W = = 3 w m (.7) Knwin the fundamental enery cnstants w m and w e, ne can determine the equilibrium micelle size frm Equatin.6 and the critical cncentratin X C frm Equatin.7. Alternatively, knwin the experimentally determined micelle areatin number and the cmc, ne can determine the fundamental enery cnstants w m and w e appearin in the Debye mdel. Debye evaluated the physical relevance f the characteristic enery parameters in his mdel by cnsiderin that fr ddecylamine hydrchlride at 5 C, the experimental cmc is 0.03 M (cnverts t mle fractin X C = ) and the areatin number is = 66. Fr these experimental values, the mdel shws, W / = 8.36 kt, w m = 5 kt, and w e =. kt. He nted that the heat f vaprizatin per ddecane mlecule is 5 kt, and that it has a crrespndence t the characteristic shrt-rane enery parameter w m. He calculated w e fr a disk cvered with a cnstant chare density in a medium f dielectric cnstant ε, t be we = 4 3 π e ε a (.8) where a is the area per head rup and e is the electrnic chare. Takin the estimate f Harkins et al. [8] that the surface area ccupied by ne head rup at the areate surface is 7 Ǻ, and assumin fr the effective dielectric cnstant (the averae f water and that f a hydrcarbn), he btained w e =.8 kt cmpared aainst. kt btained frm the experimental cmc and micelle size data..3.4 Size Variance f Equilibrium Micelle Debye als made an assessment f the distributin f micelle areatin numbers by expressin the ttal enery W f the micelle in the vicinity f = as a functin f ( ). Frm a Taylr expansin f the enery functin W arund, he btained W W W = W + W + ( ) ( ) = 3 4. (.9) This equatin is rearraned t btain the fluctuatin in the micelle areatin number = 4 W W 3 W. (.0) Intrducin natural thermal fluctuatins int this expressin, W W = kt and Equatin.6 fr W, Debye shwed that the fluctuatin in the micelle areatin number is kt = 4. (.) w m

10 One Hundred Years f Micelles Fr the experimental values f and w m determined fr the ddecylamine hydrchlride surfactant, is f the rder f 3.3, indicatin that the micelle areatin number is relatively narrwly dispersed. Debye s first theretical mdel thus intrduced the cncept f ppsin frces (ln-rane electrical versus shrt-rane van der Waals). It shwed the drivin frce fr micelle frmatin t be the attractive van der Waals interactins whereas the finite size f the micelle is determined by repulsive electrstatic frces. It identified a cmc with the recnitin that it remains practically cnstant. Frm the minimizatin f the enery f a sinle micelle, it shwed that the micelles are small. Frm the Taylr expansin analysis f the ttal enery, the mdel suested that the small micelles are narrwly dispersed in their sizes. Because the mdel was based n minimizin the enery f a sinle micelle, it can be viewed as a precursr t what later evlved as the pseudphase mdel f micelle frmatin. Hwever, it is different frm the pseudphase mdel because in a pseudphase, the enery f a mlecule that cnstitutes the pseudphase is a minimum whereas in the Debye mdel, the enery f the micelle (and nt f a sinle sap mlecule within it) is a minimum..4 evolutin OF THeRIes ADDRESSING THE SHORTCOMINGS OF THE DEBYE MODEL The pineerin thery f Debye stimulated ther theretical studies n micellizatin; sme f which attempted t imprve n the Debye thery withut questinin the basic mdel whereas thers attempted t crrect sme f the critical missins f the Debye thery. Debye had already nted that the rle f the additin f electrlyte t the surfactant slutin needed t be wrked ut. He arued that because in an electrlyte slutin, every chare will be surrunded by an excess f ins f the ppsite sin, its electrical actin will be screened ut fr larer distances. Therefre, an added electrlyte will screen the actin f the chares n the micelle, reduce the electrical wrk W e, and therefre increase the equilibrium size f the micelle. This analysis was quantitatively develped by Hbbs [], wh extended the Debye thery t a slutin cntainin electrlytes by applyin the Debye Hückel apprximatin t estimate the salt-induced chane in the inic interactin enery. He btained expressins fr the increase in the micelle size and decrease in the cmc as a functin f the added salt cncentratin. Hwever, the validity f the Debye mdel was nt questined. Hbbs als estimated the electrstatic interactins between the tw chared surfaces f the lamellar micelle. In the absence f any salt, he estimated that these interactins may accunt fr 0% f the ttal electrstatic enery and culd thus reduce the equilibrium micelle size and increase the cmc. Oshika [] and Reich [3] questined sme f the fundamental cncepts underlyin the Debye thery. In buildin a thery based n McBain s lamellar areate, Debye did nt include any enery cntributin t accunt fr the fact that the curved surface f the areate is expsed t water. The need fr addin such a surface enery cmpnent t the micelle enery was pinted ut by Oshika and Reich. Oshika shwed that it is the surface enery, rather than the van der Waals interactins between tails prpsed by Debye, which ppses the repulsive head rup interactin enery in determinin the size f the micelle. Oshika and Reich als questined the validity f minimizin the enery f a sinle micelle as a criterin fr equilibrium rather than minimizin the enery f the entire system, which wuld als include the slutin entrpy..4. Oshika Mdel Oshika [] pinted ut that in the equilibrium state, there must be areates f varius sizes and shapes, and their distributin is t be determined by statistical mechanics. Hwever, if the size (areatin number) f a micelle is lare, the mixin entrpy f the micelles in the slutin is neliible when cmpared aainst that f mnmers, and the cmc may be rearded as

11 Surfactant Science and Technly the pint where a kind f phase chane ccurs (a view recnized as the pseudphase mdel fr micelles). Therefre, the cmc is nt appreciably affected by the distributin f the micelle sizes. The finiteness f micelles at the cmc has been attributed by Debye t the cunterbalance f the van der Waals attractive enery f the hydrcarbn tails and the Culmb-repulsive enery f the plar heads. Oshika arued that in rdinary phase chanes, the phase that has a lwer free enery tends t rw infinitely because f the surface enery and nt because f the additive van der Waals enery. Hence, it is clear that ne must add t the free enery f micelle frmatin, the surface enery between the micelle and water. The surface enery f the micelle is prprtinal t the circumferential area πrh that is expsed t water, where H dentes the thickness f the lamellar areate. Because the vlume πr H f the micelle is prprtinal t the areatin number, the circumferential area πrh will be prprtinal t /. Addin this cntributin t the enery mdel f Debye, Oshika prpsed W = W e + W m + W s = 3/ w e w m + / w s (.) where w e and w m are cnstants determined by particular surfactants as defined earlier, and w S is a characteristic surface enery cnstant. T take int accunt the system entrpy cntributin t the ttal free enery, Oshika cnsidered the system cmpsed f N sap mlecules in the slutin, N sap mlecules incrprated int the micelles, and N w water mlecules. The free enery f mixin f these cmpnents in the slutin is written as F F mix mix = kt N kt N N Nw N ln + Nw ln + l N N Nw + N + Nw + N + N ln N + N w Nw + N w ln Nw + N n N w N / N + N + (.3) where k is the Bltzmann cnstant and T the abslute temperature. The three terms appearin in the first equality in Equatin.3 represent the entpic cntributins frm the mnmeric surfactant, slvent water, and micelles, respectively. Because the areatin number is appreciably larer than, the number f micelles N / will be much smaller than N and N w. Therefre, the first equality in Equatin.3 can be simplified t btain the secnd equality as shwn. Effectively, the relatively smaller cntributin frm the entrpy f micelles is nelected. Cmbinin the interactin enery (Equatin.) and the free enery f mixin (Equatin.3), Oshika btained (fr the free enery f the system) the expressin, F = kt N N ln N + N w + N Nw ln N + N w w + N W. (.4) The cmc, X C = N /(N + N w ) and the equilibrium micelle size (the areatin number ) are determined by minimizin the ttal free enery with respect t the independent variables N and, hldin the ttal sap mlecules N + N and water mlecules N w at cnstant values, respectively. One btains, W ws kt ln XC =, =. (.5) w e

12 One Hundred Years f Micelles 3 Intrducin the expressin fr W (Equatin. with = ) in Equatin.5, the cmc can be expressed as ws kt ln XC = wews wm, = (.6) w Clearly, in cntrast with the Debye thery, the equilibrium micelle size is btained frm the cunterbalance f the surface enery and the electrstatic head rup interactins. Crrespndinly, the cmc als shws an explicit dependence n the surface enery. e.4. reich Mdel The Debye and Oshika mdels, in which the electrstatic head rup interactins play a central rle, are nt applicable t nninic deterents. Reich [3] prpsed that a eneral thery f micelles shuld be develped, startin frm a treatment f nninic surfactants fr which the electrstatic interactins are nt relevant. He als criticized the Debye thery fr the same reasns as Oshika, namely, the stable micelle size must be that which results in minimum free enery fr the system and nt the wrk f frmatin f a micelle. He arued that the rwth f micelles will invlve a decrease in the ttal number f independent species (sinly dispersed surfactant mlecules and micelles) in the system, and hence will invlve a decrease in ttal system entrpy, which must be taken int accunt. Furthermre, Reich suested that the van der Waals enery f hydrcarbn chain per surfactant mlecule must increase as the micelle rws, aruin that if it remained cnstant, as in the Debye mdel, then rwth beynd a dimer wuld nt ccur. This last feature is an early recnitin by Reich f the need fr cperativity f areatin, characteristic f micelle frmatin, and is discussed in detail in Sectin.8. Reich expressed the free enery f the system described by the equilibrium relatin in Equatin. as G T S E ln X = ln K + ln X = + ln X = + ln X kt kt (.7) where ΔS and ΔE are the entrpy and enthalpy chanes assciated with micelle frmatin. Fr calculatin ΔS, he assumed a cnstant entrpy chane s per mlecule, n micelle frmatin, which is independent f the size f the micelle. T calculate the enery chane ΔE fr the nninic deterent, he cnsidered the chanes experienced by the aliphatic hydrcarbn tail f the deterent n micelle frmatin. Fr an aliphatic hydrcarbn tail in water, the mlecule will shw a tendency t fld up s that the sements f the chain can escape frm water and remain in cntact with each ther. He assumed that each hydrcarbn tail was a tihtly packed sphere, excludin any water and with a surface area A H. Of this surface area A H, a prtin a P is cvered by the plar rup, whereas the remainder, A H a P, will be the expsed hydrcarbn surface. The surface enery f this sinle mlecule is dented by ϒ. Micelle frmatin causes a fractin f the riinal expsed hydrphbic surface t be prtected frm water. This prtected fractin multiplied by ( ϒ) prvides the surface enery chane n areatin. Fr surfactant mlecules in the nnareated state, the expsed hydrcarbn surface area is (A H a P ). Fr mlecules in spherical micelles, the expsed hydrcarbn surface area is ( /3 A H a P ). Therefre, the fractin f hydrcarbn surface expsed t water that is eliminated n areatin is (A H /3 A H )/(A H a P ). Reich defined areates f the size cm at which the surface becmes cmpletely cvered with plar rups as cmplete micelles, namely, / 3 AH A A a H P H = AH A a H P ( ) = = / 3 at cm. (.8)

13 4 Surfactant Science and Technly If an areate rew larer than the cmplete micelle size and still remained spherical, the surface wuld nt be able t accmmdate all the plar rups. Sme plar rups wuld have t be buried in the interir (this is unlikely eneretically). Such lare areates wuld presumably be flattened sufficiently t create enuh extra surface t accmmdate all the plar rups. Thus, as increases beynd the cmplete micelle size cm, the areates becme larer and flatter. The fractin f the hydrcarbn surface eliminated remains at. Accrdinly, fr values f less than that f the cmplete micelle, Equatin.7 becmes ln X = AH Ts + Υ A a H P kt / 3 + ln X (.9) whereas fr values exceedin that f the cmplete micelle (fractinal surface remved is in this case), it becmes ln X Ts + Υ = + ln X (.0) kt Reich was able t calculate and plt a micelle size distributin based n the abve equatin fr a nninic surfactant. T estimate the entrpy chane s per mlecule, he calculated the standard entrpy chane fr cndensin a hydrcarbn mlecule frm vapr state t liquid state, hldin the density cnstant. He recnized that there wuld be sme cntributin frm the entrpy chane assciated with the plyxyethylene type head rup f the surfactant, althuh it culd nt be estimated. He calculated the enery parameter ϒ frm cnsideratin f the macrscpic interfacial free enery fr a hydrcarbn. The calculated size distributin functin (Fiure.4) shwed (i) hw the cncentratins f the micelles f sizes different frm the cmplete micelle fall ff sharply; (ii) hw the chemical ptential f the surfactant, r explicitly, the cncentratin f the unareated surfactant cntrls the micelle X = X = X = ln X Areatin number FIGURE.4 Size distributin f micelles calculated by the Reich mdel fr a nninic deterent at different mnmer cncentratins. Parameters used in the cmputatins are A H = 4a P, s = 0 k, and ϒ = 30 kt, all taken frm the article by Reich. (Frm I. Reich, J. Phys. Chem., 60, 57 6, 956.)

14 One Hundred Years f Micelles 5 size distributin; (iii) the sharpness f the cmc and the narrw size distributin f the micelles; and (iv) that the size f the cmplete micelle depends n the rati a P /A H. All these features are qualitatively valid fr what we knw t be the prperties f nninic deterents. Hwever, because f the definitin f the cmplete micelle, the mst prbable micelle size always remained that f the cmplete micelle withut bein affected by the ttal surfactant cncentratin. Reich als discussed the prblem f the McBain micelle. In this case, the micelle is assumed t rw in tw dimensins, with the hydrcarbn chains alined parallel t each ther and the plar rups cverin the flat faces. Irrespective f the size f such an areate, the hydrcarbn water interface will never be eliminated cmpletely because the deterent mlecules at the ede will have their hydrcarbn chains expsed t water. In develpin an expressin fr the enery chane in the McBain micelle, which fllws similar cnsideratins as the spherical micelle, Reich shwed that the frmatin f the McBain micelle wuld indeed crrespnd t a cnditin f phase separatin, a cnclusin that we currently understand fr lamellar areate shapes..5 emergence OF A THEORY FOR RODLIKE MICELLES Early discussins n micelle shapes did nt include cnsideratin f the rdlike micelles as we recnize them presently. Philippff [7] analyzed smtic activity, specific cnductivity, and x-ray data btained at varius electrlyte and surfactant cncentratins and fr varius surfactant tail lenths and identified tw reimes with respect t micellar shapes. In the first reime, which crrespnds t zer r a small amunt f electrlyte, micelles are practically spherical in the sense that they need nt be true spheres but can be cubes, shrt cylinders, prisms, r spherids. In the ther reime, which crrespnds t lare electrlyte cncentratins and lner tail lenths f surfactants, the areates are anismetric and lare. Philippff cnsidered that it is imprbable fr small micelles t reranize int lare micelles with increasin salt and surfactant cncentratins and therefre pstulated the lare structures t be secndary areates f the prefrmed primary micelles. The frmatin f rdlike micelles as we understand them currently was first prpsed by Debye and Anacker [4] based n liht scatterin measurements n slutins f catinic alkyl trimethyl ammnium brmides in the presence f added salt, KBr. They cncluded that the dissymmetry measurements were nt in areement with the frmatin f spherical micelles and cylindrical lamellar micelles but areed with micelles bein very lare and rdlike at hih cncentratins f added salt. They prpsed that the crss-sectin f the rd wuld be circular with the plar heads f the deterent lyin n the periphery and the hydrcarbn tails fillin the interir. The ends f such a rd wuld mst certainly have t be runded ff with plar heads. They bserved that the effect f salt n micelle rwth was mre sinificant, the lner the tail lenth f the surfactant. A number f studies that fllwed cnfirmed the frmatin f such lare rdlike micelles, as summarized in the article by Anacker [4]. The plydispersed nature f rdlike micelles was first recnized by Scheraa and Backus [43]. They cnducted flw birefrinence measurements in slutins f cetyltrimethylammnium brmide (CTAB) and cncluded that lare asymmetrical areates are frmed whse lenth increases with increasin salt cncentratin. Mre imprtantly, they fund that a mndispersed areate mdel des nt describe the experimental data and that areates must be hihly plydispersed. They prvided an interpretatin fr this size distributin by aruin that fr spherical micelles frmed at lw salt cncentratins, relatively hih eneries are invlved (based n Debye s thery) if the micelle size is t be chaned by mre than tw r three mnmers. Hence, the small micelles shuld have narrw size distributins. In cntrast, at hih salt cncentratins, the chared rups becme shielded s that it des nt require much wrk t be dne t brin up mre mlecules. This wuld increase the mean micelle size and als lead t a hih plydispersity. They als speculated as t the bendin and flexibility f the micelles, althuh these features were nt explred.

15 6 Surfactant Science and Technly The existence f viscelastic behavir in surfactant slutins was first discvered by Pilpel [44,45], based n rhelical studies n aqueus slutins f sdium and ptassium leate in the presence f electrlytes. He interpreted the viscelasticity as arisin frm a chane in shape f the areates, with the small sap micelles frmed at lw electrlyte cncentratins transfrmin t ln interlinked cylinders at hih electrlyte cncentratins..5. halsey Mdel The first theretical cncept t describe rdlike micelles was prpsed by Halsey [4]. As discussed previusly, tw factrs that limit the rwth f micelles and permit them t exist withut phase separatin had been prpsed: ne is the emetrical limit f chain lenth prpsed by Hartley as the limitin radius f spherical micelles and the ther is the repulsive interactins between the head rups at the micelle surface prpsed in the Debye mdel (applied t the McBain Harkins cylindrical prtin f a lamellar micelle). Halsey [4] examined the rwth f micelles avidin bth f these limitatins, which resulted in finite stable areates withut causin phase separatin. Usin Debye s apprach, Halsey cmpared the eneretics f ne-dimensinal rwth f micelle as a disk and as a rd. Halsey calculated the electrical ptential at a small distance z frm the end f a ln rd f lenth L = nz, where z is a characteristic distance between chares and w e is the characteristic electrical enery appearin in the Debye mdel fr electrstatic head rup interactins. Fr lare n, the ptential will be w e nz dz = we ln n. (.) z z The ttal electrical wrk f frmin a cylindrical micelle f size is iven by We = we ln n dn = we[ ln + ]. (.) Cmparin this expressin fr electrical enery f cylinders with the Debye expressin (Equatin.4) W e = w e 3/, fr lamella, Halsey nted that the functin ( ln + ) varies mre slwly than 3/ as increases. Therefre, he cncluded that fr lare, a rdlike micelle will crrespnd t lwer enery and hence the mre stable areate cmpared with a disk. He arued that because the surfactant mlecules apprachin ne end f a ln rd wuld feel nly the near end, the rd lenth shuld have n effect f n the ttal enery. This wuld make the lenth f the rd t be infinite. Because this is nt the case, he prpsed that the finite size f the micelles can be explained by the analy between rdlike micelles and a ne-dimensinal as (mlecules fixed n a strin like beads). Just as a ne-dimensinal as des nt cndense whatever may be the nature f interactins, the rdlike micelles remain finite and d nt cause phase separatin. The micelles can be plydispersed analus t the distributin f a linear plymer in equilibrium with the mnmer. As fr the ther tw smaller dimensins f the rdlike micelle, Halsey bserved that ne dimensin must be limited by the lenth f the hydrcarbn chain and the secnd can either be limited by electrical frces (as in the Debye mdel) r by the lenth f the hydrcarbn chain (based n the structural descriptin prpsed by Hartley). Halesey s cnclusins as t the lare micelles bein cylindrical, finite in size, plydispersed, and that such micelle frmatin is different frm phase separatin remain valid tday.

16 One Hundred Years f Micelles 7.6 statistical MECHANICAL THEORIES OF MICELLES With the develpment f fundamental statistical mechanical mdels describin liquid state, such frmalisms were als applied t describe the frmatin f micelles. Tw f the sinificant cntributins are due t Heve and Bensn [5] and Pland and Scheraa [6,7] because, in bth f their mdels, the free enery f frmatin f micelles was decmpsed int many individual cntributins identified in mlecular terms similar t mre recent predictive theries. These studies mre clearly enumerated the varius factrs cntributin t the free enery f micellizatin, includin the hydrphbic effect assciated with the transfer f surfactant tail frm water t micelle and the surface enery assciated with the expsed hydrphbic surface f the micelle..6. heve and Bensn Thery In Heve and Bensn s treatment [5], the micelles are thuht t be areates f the hydrcarbn parts f the mlecules, with the plar parts n the utside f the areate in cntact with water. It was assumed that a spherical shape persists until the micelle becmes lare enuh that the radius f the sphere wuld exceed the maximum lenth f the hydrcarbn part f the mlecule. When that ccurs, the micelle is assumed t becme flatter, leavin the plar parts fully hydrated and an blate sphercylinder shape was assumed. The interir f the micelle is assumed t have prperties similar t that f a liquid hydrcarbn. It is assumed that the system may be treated by classic partitin functins, apart frm cntributins f vibratins, which are cnsidered t be quantized and separable. Areates f all sizes are assumed t exist, with N bein the number f areates f size. Fr slutins f nninic deterents in water, the mst prbable micelle size distributin is determined by minimizin the free enery F r maximizin the cannical partitin functin Q f the slutin (nte, F = kt ln Q). Heve and Bensn btained Q V Q V ln N = ln, ln N = λ Λ ln Λ + λ ln N QV Q V = ln N ln Λ Λ + ln 3 3 (.3) where V is the system vlume ver which all the areates translate, Q and Q are the internal partitin functins fr the unareated mlecule and an areate f size, Λ and Λ are factrs appearin in the external translatinal/rtatinal partitin functins fr the mnmer and areate, and λ is the Larane multiplier that links the tw equatins prvidin an expressin fr the size distributin f areates. The translatinal/rtatinal terms Λ and Λ are iven by / h h Λ = Λ Λ = trans rt πmkt πikt / = = trans rt h h Λ Λ Λ πmkt πi kt / / (.4) where m is the mass f a surfactant mlecule, h is the Planck cnstant, and I and I are the averae mments f inertia f a mnmer and areate f size, respectively. T evaluate Q fr the areate, multiple cntributins were cnsidered. First, the assumptin that the interir f a micelle is liquid in character was used. Heve and Bensn suested that

17 8 Surfactant Science and Technly because paraffinic hydrcarbns are nt appreciably curled up in the liquid state, the cntributins t the partitin functin f a hydrcarbn liquid are separable, and they wrte dwn the fllwin liquid state cntributin t Q. Q = 3 πmkt πihkt Vf h h! / 3/ 8π γ q h (.5) Here, I h is the mment f inertia f the hydrcarbn chain, q h is the vibratinal partitin functin fr the chain, V f is the free vlume within the liquid phase where mlecular translatin ccurs, and γ is the rtatinal free anle rati accuntin fr the hindered rtatin in the hydrcarbn liquid. A secnd cntributin cmes frm the recnitin that the areate is a small liquid drp and therefre has a surface cntributin. Fr a spherical drp f mlecules, with each mlecule havin a surface area A H, the drp surface area is /3 A H. The surface area A H is calculated frm the radius r f a hypthetical sphere whse vlume is identical t that f the surfactant tail. Heve and Bensn prpsed the surface cntributin as the prduct f the surface area and a characteristic interfacial enery σ between the drp and water. Q exp A / 3 σ exp 4πr / 3 σ (.6) = ( H ) = ( ) Heve and Bensn then cnsidered a third cntributin t accunt fr the cnstraint that in the micelle all ends f the hydrcarbn chains t which the plar heads are attached must be at the surface f the micelle, whereas in the liquid drp, the crrespndin ends may ccupy any psitin within the drp. The cntributin f brinin the chain ends t the surface, resultin in head rup crwdin, was apprximated as fllws. Befre brinin the ends t the surface, the free vlume available fr them is prprtinal t the micellar cre vlume V = v, with v dentin the vlume f a surfactant tail. After brinin the ends t the surface, the free vlume is prprtinal t the free surface area available fr them, which is assumed t be the difference between the surface area f the spherical micelle A and the space a P ccupied by the head rups already at the surface, that is, A a P. Cnsequently, this cntributin was written as prprtinal t the free vlume restrictin n micelle frmatin. Q C A a crwd P = V (.7) Finally, a furth cntributin was recnized fr nninic surfactants with plar parts that are ln plar chains (such as in the mst cmmn nninic surfactant family with liethylenexide head rups), which strnly interact with the slvent and pssess internal derees f freedm. Usin q P, they dented the cntributin f the plar head due t internal vibratins and rtatins and interactins with the slvent. Therefre, Q head Cmbinin all f these cntributins, ne can write = ( q ) (.8) P Q = Q liq Q sur Q crwd Q head. (.9)

18 One Hundred Years f Micelles 9 Fllwin the same apprach as fr the micelle, Heve and Bensn cnsidered an expressin fr the partitin functin Q f the mnmer. In evaluatin Q, the cntributins frm the plar part and the hydrphbic chain were separated, which is similar t what was dne fr the areate. Because the plar parts are assumed t remain in the water phase, q P has the same cntributin per mlecule fr micelles and fr the sinle mlecule. Similarly, the internal vibratinal cntributin q h is als the same fr the mnmer as fr the mlecule in an areate. Hwever, the cntributin f the hydrcarbn part interactin with the surrundin water culd nt even be apprximately evaluated. They recnized that structural effects exist in the water surrundin the hydrcarbn parts f the sinle mlecules, and that the mdel f a liquid drp cannt be valid here. It was felt that nly after ainin a better understandin f these effects culd the cntributin f the hydrcarbn part be quantitatively btained. Keepin it as an unknwn mlecular partitin functin q w, the verall partitin functin f the mnmer was written as Q = q w q h q P. (.30) By intrducin the partitin functins (Equatins.9 and.30) in the expressin fr areate size distributin (Equatin.3), Heve and Bensn were able t identify sme eneral features f the micellizatin f nninic surfactants. The illustrative size distributin curves they calculated fr tw different mnmer cncentratins are shwn in Fiure.5. The size distributin functin was calculated shwin a minimum and maximum in the size distributin and withut the piecewise cntinuity intrduced by Reich because f his pstulate f a cmplete micelle. As a result, the maximum in the size distributin culd be seen t chane with increasin surfactant cncentratin, althuh nly a little, and nt fixed t ne micelle size ( cm f the cmplete micelle) as in the Reich mdel. The cmc phenmenn is als evident because the amunt f surfactant incrprated int the areates (excludin mnmers) increases frm apprximately 0 8 t and. 0 4 fr the three mnmer cncentratins shwn in Fiure.5. A small decrease in the mnmer cncentratin clse t the cmc makes the areate cncentratin clse t zer. A small increase in the mnmer cncentratin clse t the cmc causes an rder f manitude increase in the amunt f surfactants present as areates. The wrk f Heve and Bensn is imprtant fr many reasns. It was the first frmal applicatin f the statistical mechanical frmulatin t micelle frmatin, even thuh the need fr such X = X = X = ln X Areatin number FIGURE.5 Size distributin f micelles calculated usin the Heve and Bensn mdel fr a nninic deterent at three different mnmer cncentratins. The parameters fr the mlecule used in the cmputatins are identical t thse used in the article by Heve and Bensn [5].

19 0 Surfactant Science and Technly an applicatin was freseen in the wrk f Oshika, Reich, and Halsey. The frmulatin required the identificatin f multiple cntributins t the partitin functin (r the free enery) and three imprtant new cntributins were identified cmpared with all previus studies that had identified van der Waals, electrical, and surface cntributins. First, the treatment identified the imprtance f cnsiderin the mlecular translatinal and rtatinal cntributins, and these are part f the chain packin cntributins we cnsider in current predictive theries. Secnd, by accuntin fr the crwdin f head rups at the micelle surface fr the nninic surfactants, Heve and Bensn were the first t prpse a clear free enery cntributin respnsible fr the finiteness f micelle size fr nninic surfactants. Third, Heve and Bensn were the first t explicitly recnize that water structural chanes will prvide a critical cntributin t the free enery and a mre fundamental understandin f these structural chanes is required befre a quantitative predictive mdel f micelles can be develped. The subsequent wrk f Pland and Scheraa [6,7] and f Tanfrd [8 30] indeed emphasized detailed structural descriptins f these water structure chanes and the quantitative and accurate estimatin f the crrespndin hydrphbic free enery. Cncernin inic micelles, Heve and Bensn nted that the micelles are hihly chared and therefre the Debye Hückel apprximatin is nt adequate t describe inic micelles. In applyin the Pissn Bltzmann equatin, the challene was t accunt fr the fact that the cunterins in the neihbrhd f the micelle are crwded, even if the slutin has a lw cncentratin f ins. Anther difficulty they pinted ut related t the questin f hw ruh the micelle surface is. Because the cncentratin f cunterins is quite lare in this rein, the electrical free enery wuld be rather sensitive t the deree f ruhness f the surface. These features relevant t the estimatin f electrstatic interactins at the micelle surface remain unsatisfactrily explred even tday and limit ur ability t accurately predict areatin prperties in the presence f a variety f cunterins, especially the cunterin specificity in prmtin a transitin frm spherical t rdlike micelles..6. Pland and Scheraa Thery Pland and Scheraa [6,7] undertk the mdelin f micelle frmatin as an illustratin f the hydrphbic effect fr which Nemethy and Scheraa [46 48] had develped a quantitative thery just a few years earlier. T simplify the calculatins, instead f cnsiderin a size distributin f micelles, Pland and Scheraa [6] cnsidered a system cntainin N surfactant mlecules, present in the frm f N micelles f size. The equilibrium prperties f the micelle were determined frm the minimizatin f the free enery F() f the system with respect t the micelle areatin number. Pland and Scheraa cnsidered spherical micelles with plar heads n the surface and nnplar tails in the interir partially ciled up (with sme freedm f internal mtin) and interactin with each ther thruh hydrphbic bnds. The free enery f frmatin f these bnds was quantitatively estimated frm the thery f hydrphbic bndin develped by Nemethy and Scheraa. They cnstructed a partitin functin Q fr the micelle with cnstituent mlecules, by identifyin an external cntributin, an internal cntributin, and a slvent interactin cntributin. F( ) = kt ln Q N N,! Q = Q Q Q ext int sl ext F( ) eq eq = ln = ln + lnq + lnq NkT N N int sl. (.3) The external cntributin is taken as the prduct f the classic partitin functins fr the translatin and rtatin f the micelle as a whle, similar t that in Heve and Bensn s thery.

20 One Hundred Years f Micelles Q ext mkt IhkT = 3 Vf π π h h / 3/ 8π γ (.3) Here, I h is the mment f inertia f the hydrcarbn chain, q h is the vibratinal partitin functin fr the chain, V f is the free vlume within the liquid phase where mlecular translatin ccurs, and γ is the rtatinal free anle rati accuntin fr the hindered rtatin in the hydrcarbn liquid. The internal partitin functin crrespnds t the internal freedm f the micelle arisin frm the mtins f the hydrcarbn tails. A mnmer mlecule in a micelle will make tw lare cntributins t the internal partitin functin. One is the mtin f the mnmer mlecule as a whle within the micelle and the ther is the internal rtatin (includin cmplex vibratins and trsinal scillatins) in the hydrcarbn tails. Because the latter is included in the hydrphbic bnd enery per mlecule and is accunted fr as part f the slvent cntributin discussed belw, nly the cntributin f the frmer had t be included in Q int. T accunt fr the internal mtin f the mnmer in the micelle, they cnsidered tw appraches. In ne, a free vlume viewpint is taken and the internal mtin is treated as a translatin; in the ther, the micelle is treated as a lattice and the permutatins f mnmers in the lattice are calculated. Frm bth appraches, they btained essentially the same dependence f Q int n. Takin the free vlume apprach, Q int = πmkt h! 3/ V, V = V f f sh. (.33) In Equatin.33, V sh is the vlume f a spherical shell within which all the head rups are cnstrained t translate and the factr ( )/ represents the fact that the vlume ccupied by a mlecule is nt available t it fr translatin. Fr calculatin the slvent interactin partitin functin, they applied the results frm the thery f hydrphbic bndin develped by Nemethy and Scheraa. In the mnmeric state, the entire surface area f the surfactant tails cmprises the hydrphbic surface expsed t water. In the micellar state, nly the surface f the spherical micelle, excludin the space ccupied by the head rups, cmprises the hydrphbic surface expsed t water. Therefre, ne can calculate the fractin θ H f the ttal hydrcarbn surface invlved in hydrphbic bndin, similar t that dne by Reich. Pland and Scheraa expressed the slvent interactin cntributin as the prduct f this fractinal area f expsure θ H invlve in hydrph ic bndin and the hydrphbic bnd enery ΔF H per mlecule. The hydrphbic bnd enery pe mlecule had been estimated frm Nemethy and Scheraa s thery f hydrphbic bndin as a functin f temperature and the chain lenth f the mlecule. The slvent interactin partitin functin is iven by Equatin.34. Here, fr the hydrphbic bnd enery, they used a temperature-dependent expressin (which they established fr amin acids) with the cnstants c, c, and c 3 taken t crrespnd t alanine alanine bnds. The fractinal area f expsure is apprximated in the last step by nelectin the area a P cvered by the head rup in cmparisn with the surface area A H f a tail. Q θ H FH = exp H FH c c T c T θ RT, = / 3 AH A = A a H P H AH = AH ap (.34) / 3 / 3.

21 Surfactant Science and Technly Pland and Scheraa [7] als extended their treatment t inic surfactants by includin an electrstatic free enery fr spherical micelles. They calculated this free enery by cnsiderin the Culmbic interactins between the chared head rups similar t the mdel f Debye, but applied t a spherical surface f radius R. As mentined previusly, the radius R is related t the areatin number thruh the relatin R = r /3, where r dentes the radius f a spherical hydrcarbn drplet whse vlume is equal t that f a sinle surfactant tail. Therefre, the electrstatic partitin functin was represented as Q ele e e = exp = exp / εr εr 3. (.35) Cmbinin the different free enery cntributins, the verall dependence f the free enery F() n the areatin number has the frm F( ) NkT FH = ( α ln X + 5 ln ) + RT / 3 ( ) β + / 3 ln + γ (.36) 3 where α is extracted frm the external free enery term (Equatin.3), X is the surfactant cncentratin, β is extracted frm the internal free enery term (Equatin.33), and γ is extracted frm the electrstatic enery term (Equatin.35). Nte that β and γ are relevant nly fr the micelle and will nt appear in the free enery f a mnmer. Pland and Scheraa specified three cnditins t btain a stable micelle in a system cntainin N surfactant mlecules: (C) the free enery F() f the slutin f micelles must be an extremum ( F/ = 0), (C) the extremum must be a minimum ( F/ < 0), and (C3) the free enery f the system f micelles F() shuld be lwer than the free enery f the system f mnmers, F(). These three cnditins lead t the fllwin three cnstraints, respectively, n the hydrphbic bnd enery, required fr the frmatin f micelle f size, where the last cnditin has been apprximated takin int accunt that is much larer than. Furthermre, if the third cnditin is made int an equality, it prvides the relatin fr the cmc. F H / 3 3 C : 0 ( ln = F = α X + + RT 5 ln 5) + γ F < 0 C FH 3 / 3 : RT < γ C3 : F( ) < F( ) FH ln > + α X ln β RT 3 + γ. / 3 (.37) The cnditins fr micelle frmatin based n the abve equilibrium requirements are shwn in Fiure.6 as a relatin between the hydrphbic bnd enery and the micelle size. The first cnstraint in Equatin.37 implies that, fr a iven cncentratin, micelles will be frmed when the values f ΔF H and lie n the curve C. The secnd cnstraint implies that micelles will frm nly fr ΔF H and values, which lie belw the curve C. Because the surfactant cncentratin fixes the curve C, curves C and C determine the values f the hydrphbic bnd enery and the crrespndin micelle size. The secnd cnstraint impses a restrictin n the manitude f ΔF H fr micelle frmatin t ccur. If the hydrphbic bnd strenth is t lare, small limers wuld be preferred t micelles. Fr lwer hydrphbic bnd enery, the frmatin f small limers is

22 One Hundred Years f Micelles 3 (a) 0 C - Nninic C - Nninic C - Inic C - Inic F H /RT (b) F H /RT Areatin number 0 9 C C (X = 0 4 ) C (X = 0 3 ) C (X = 0 ) Areatin number FIGURE.6 (a) Pland and Scheraa mdel predictins f cnditins fr micelle frmatin frm inic and nninic surfactants at a surfactant cncentratin f X = 0 3. Curves C and C are calculated usin Equatin.37 with the assumed mdel parameters f α =, β = 0., and γ = Micelle frmatin is allwed nly fr cnditins belw the curve C. Equilibrium micelle size at the iven cncentratin X can be fund frm the line C dependin n the value f the hydrphbic bnd enery f the surfactant. The leftward shift in the intersectin f the curves C and C fr the inic surfactant imply that the equilibrium micelle size fr the inic surfactant will be smaller than that fr the nninic surfactant. (b) Shift in micelle size with a chane in surfactant cncentratin fr inic surfactants. Curve C is calculated fr three surfactant cncentratins whereas curve C is independent f cncentratin. The arrws indicate the predicted equilibrium micelle size fr the three cncentratins, crrespndin t the hydrphbic bnd enery f 9. units. The micelle size increases with increasin surfactant cncentratin. disallwed. Cnditin C3 indicates whether the cncentratin fr which the curve C is cnstructed is abve r belw the cmc. In the Pland and Scheraa mdel, the external, internal, and slvent interactin free enery cntributins fr the nninic surfactants vary radually when the areatin number exceeds that f small limers, say 0. Crrespndinly, the minimum in the free enery fr nninic surfactants is shallw resultin frm a delicate balance between the slwly varyin free enery cntributins. In cntrast, fr inic surfactants, the minimum is sharper because f the strner dependence f the electrstatic enery n the micelle areatin number. In the Pland and Scheraa treatment, the

23 4 Surfactant Science and Technly hydrphbic effect was fully recnized and the first attempt t emply a reasnable quantitative estimate was made. The residual interfacial cntact between the hydrcarbn tail and water was accunted fr in the slvent interactin term as in the Reich mdel rather than treatin it as a surface free enery as was dne in the Heve and Bensn thery..7 MICELLE SHAPE TRAnsITIns AND SIZE DIstRIBUTIn The free enery mdels cnstructed thus far have fcused mainly n small micelles havin either the lamellar-cylinder (McBain Harkins micelle) r spherical (Hartley micelle) shape. Subsequent develpments in the thery f micelle frmatin have attempted t cnstruct a unified free enery mdel fr the cmmn areate shapes bserved in surfactant slutins (Fiure.7). The small micelles are spherical in shape. When lare rdlike micelles frm, they can be visualized as havin a cylindrical middle prtin and parts f spheres as endcaps. The cylindrical middle and the spherical endcaps can have different diameters. When micelles can n lner pack int spheres (this happens fr areatin numbers fr which a spherical areate will have a radius larer than the extended lenth f the surfactant tail), and if at the same time the rdlike micelles are nt yet favred by equilibrium cnsideratins, then small nnspherical lbular areates frm. Glbular shapes such as prlate and blate ellipsids and shapes enerated via ellipses f revlutin have been prpsed fr these micelles. Sme surfactants pack int a spherical bilayer structure called a vesicle, which enclses an aqueus cavity. In the uter and the inner layers f the vesicle, the surface area (in cntact with water) per surfactant mlecule and the number f surfactant mlecules need nt be equal t ne anther and the thicknesses f the inner and uter layers f the bilayer can als be different frm ne anther. The multiple equilibrium mdel f micellizatin (Equatin.) can be frmally applied t areates f any shape and size. Crrespndinly, the equilibrium cnditin crrespndin t the minimum f the free enery f a slutin made up f mnmers, areates f all sizes and shapes, and water mlecules can be represented in the frm shwn in Equatin.38, which stipulates that the chemical ptential f the sinly dispersed surfactant mlecule is equal t the chemical ptential per mlecule f an areate f any size and shape. In the multiple equilibrium descriptin, each areate f a iven size and shape is treated as a distinct chemical cmpnent characterized by a chemical ptential. The cnditins crrespndin t the frmatin f areates are usually in the Spherical micelle Glbular micelle Rdlike micelle Spherical bilayer vesicle FIGURE.7 Schematic representatin f surfactant areates in dilute aqueus slutins. The structures frmed include spherical micelles, lbular micelles, rdlike micelles with spherical endcaps, and spherical bilayer vesicles. One characteristic dimensin in each f these areates is limited by the lenth f the surfactant tail.

24 One Hundred Years f Micelles 5 realm f dilute slutins and, fr these cnditins, ne can very simply relate the chemical ptential f a cmpnent t the cncentratin f that cmpnent in slutin, as shwn in Equatin.38. µ = µ µ = µ + kt ln X, µ = µ + kt ln X. (.38) Here, µ is the standard state chemical ptential f the areate f size havin any shape, X is its mle fractin in slutin, k is the Bltzmann cnstant, and T is the abslute temperature. The standard state f the slvent is defined as the pure slvent whereas the standard states f all the ther species are taken t be infinitely dilute slutin cnditins. Cmbinin the equilibrium cnditin with the cncentratin dependency f chemical ptentials, we btain the areate size distributin X = X exp X kt = exp µ µ µ kt. (.39) Here, µ is the difference in the standard chemical ptentials between a surfactant mlecule present in an areate f size and a sinly dispersed surfactant in water. It is this free enery difference that is directly cnnected t the equilibrium cnstant K fr micellizatin defined in Equatin.. T calculate the areate size distributin, we need an explicit equatin fr the standard state chemical ptential difference µ r equivalently, the equilibrium cnstant fr areatin, K. Mst f the theretical studies in the last 30 years have fcused n develpin quantitatively accurate expressins fr the dependence f this equilibrium cnstant n. Even in the absence f a free enery mdel fr micellizatin, the thermdynamic relatins prvide many interestin results pertinent t the self-assembly prcess [49]. Frm the micelle size distributin, we can cmpute averae areatin numbers usin the definitins 3 X X n =, w =, z = X X X X (.40) where n, w, and z dente the number-averae, the weiht-averae, and the z-averae areatin numbers, respectively, and the summatins extend frm t. The ratis w / n and z / w are unity fr mndispersed systems and are equal t and 3/ fr systems exhibitin very hih plydispersity. Thus, either f these ratis can be used as an index f plydispersity. Fr a surfactant with any kind f head rup, we can shw [49] frm the size distributin that the averae areatin numbers n and w depend n the cncentratin f the micellized surfactant (difference between the ttal surfactant cncentratin X T and the cmc X C ) as fllws: n = = z ln n ln X, ln w ln n w n w ( X ), w ( X ) z w w X, X = X X. T C (.4) This equatin states that the averae areatin numbers n and w must increase appreciably with increasin cncentratin f the micellized surfactant if the micelles are plydispersed; the

25 6 Surfactant Science and Technly averae areatin numbers must be virtually independent f the ttal surfactant cncentratin if the micelles are narrwly dispersed. Furthermre, Equatin.4 shws that the expnent relatin the averae micelle size t the ttal surfactant cncentratin is a direct measure f the areate plydispersity. These are purely thermdynamic results independent f any free enery mdels fr micellizatin. The cnclusins f Debye as t the narrw distributin f small micelles and f Halsey n cylindrical micelles bein plydispersed are cnsistent with these thermdynamic results, which are independent f any free enery mdel..8 CPERATIVE AND AntICPERATIVE FREE ENERGY FUNCTIn Utilizin an increasin understandin f the eneretic factrs cntributin t the micellizatin prcess, particularly with the recnitin f hydrphbic interactins, Mukerjee was able t shw hw ne part f the free enery functin must prmte the rwth f the areate (the cperative part) and anther part f the free enery functin must limit the rwth and cntribute t the finiteness f the areate (the anticperative part) [36,50 53]. In develpin his analysis, Mukerjee als discvered hw very subtle chanes in the tw parts f the free enery functin can affect the frmatin f spherical micelles versus rdlike micelles. As previusly mentined, Reich had already recnized the need fr cperativity in micelle frmatin fr a micelle f sme lare enuh areatin number t frm whereas the frmatin f dimers, trimers, and ther small limers is prevented..8. representatin f Cperativity Mukerjee started with the representatin f micelle frmatin as a stepwise assciatin prcess. A + A A, K = k (.4) k j where, k is the stepwise assciatin equilibrium cnstant fr the frmatin f a -mer frm the cmbinatin f a ( )-mer with a mnmer. It is differentiated frm the mnmer micelle equilibrium cnstant K defined in Equatin. and is related t it as shwn abve. The stepwise assciatin equilibrium cnstant is als directly linked t the free enery chane µ thruh the relatin d( µ ) kt lnk = µ ( ) µ = ; kt lnk = µ. (.43) d Thus, the verall mnmer micelle equilibrium cnstant K is directly related t the manitude f the free enery f micellizatin µ whereas the stepwise assciatin equilibrium cnstant k is related t the dependence f this free enery n. If k increases with, then the larer areates are favred ver the smaller nes and the system is cnsidered t exhibit psitive cperativity. If k decreases with, then the frmatin f larer areates is increasinly disfavred and the system is said t exhibit neative r anticperativity. When k is independent f, the assciatin is said t be cntinuus and nncperative. In this case, plydispersed areates frm and their size distributin is mntnically decreasin. The shieldin f the hydrphbic part f the micellar cre frm water becmes mre and mre effective with every incremental additin f a surfactant t the micelle. Thus, the incremental chane in free enery due t the hydrphbic interactins becmes mre neative with increasin size f the micelles. This has the tendency t increase k with increasin. Hwever, as increases, the micellar surface becmes increasinly crwded with the plar head rups f the surfactants.

26 One Hundred Years f Micelles 7 Cnsequently, fr every incremental additin f a surfactant t the micelle, the repulsin between the plar head rups increases. This has the tendency t decrease k with increasin. At the initial staes f areatin, that is, fr relatively small values f, the incremental chane in the hydrphbic interactins is reater than that in the head rup repulsins. Hence, there is an initial rein f psitive cperativity in which k increases with increasin. Beynd sme critical areatin number, the incremental chane in the head rup repulsins exceeds that in the hydrphbic interactins. Therefre, beynd a maximum value crrespndin t a critical areatin number, k beins t decrease with, sinalin a rein f neative cperativity. When lare cylindrical micelles bein t frm, the incremental chane in the surface area f the micelle per amphiphile becmes a cnstant. As a result, the incremental additin f a surfactant mlecule t the micelle alters neither the incremental chanes in the attractive hydrphbic interactins nr the repulsive head rup interactins. Cnsequently, k becmes independent f, indicatin a final rein f nncperativity..8. cperativity and Frmatin f Cylindrical Micelles Mukerjee [36,5] prpsed an empirical equatin (Equatin.44) fr the functinal dependence f K and k n t shw that subtle chanes in the anticperative rein determine whether small micelles r lare cylindrical micelles frm. µ lnk = = ( )ln( ) 0. 0 ( ) ( ) kt d( µ / kt ) lnk = = ln( ) ( ) d (.44) Specifically, he cnsidered tw situatins differin frm ne anther in the value f where the anticperative rein ends and the nncperative rein beins, as shwn in Fiure.8. Mukerjee shwed that when the anticperative rein extends t the areatin number 6 and beynd, nly small spherical r lbular micelles with narrw size distributin frm. The averae size des nt k = k 6 fr > 6 k = k 96 fr > 96 0 ln k Areatin number FIGURE.8 Schematic representatin f the stepwise assciatin equilibrium cnstant, shwin the rein f cperativity and the rein f anticperativity. At lare areatin numbers, when cylindrical micelles frm, the stepwise assciatin cnstant becmes independent f size and is a cnstant, crrespndin t a rein f nncperativity. The stepwise assciatin equilibrium cnstant shwn here is calculated usin the empirical Equatin.44 used by Mukerjee [36,5] t illustrate the phenmena.

27 8 Surfactant Science and Technly chane even when the ttal surfactant cncentratin is increased by rders f manitude. In this case, the shallw free enery minimum and its rane (determined by where the anticperative rein ends) are favrable enuh fr the stability f the smaller micelles and a transitin t rds des nt ccur. On the ther hand, when the anticperative rein ends at an areatin number f 96, lare plydispersed rdlike micelles frm, which sinificantly chane their averae size with increasin surfactant cncentratin. In this case, the rane f the shallw free enery minimum is nt lare enuh t assure the stability f the small micelles and a transitin t the lare micelles ccurs. Mukerjee [50,5] was the first t treat the thermdynamics f rdlike areates by recnizin that tw characteristic equilibrium cnstants are necessary t describe their frmatin. In the stepwise areatin prcess, Mukerjee emplyed an equilibrium cnstant k fr the frmatin f a dimer that was different frm the equilibrium cnstant k fr the subsequent stepwise assciatin fr all areates larer than the dimer. Frm the equilibrium relatins, Mukerjee shwed hw the weiht averae areatin number w f the micelle is related t the cncentratin f the surfactant in slutin and als the dependence f the cmc (X C ) n the equilibrium cnstants, as fllws: w k = k / / X T X X XC X, =. k (.45).8.3 alternate Representatins f Cylindrical Micelles Frmatin The apprach pineered by Mukerjee is equivalent t the thermdynamic treatments presented in later studies by Tausk and Overbeek [54], Israelachvili et al. [3], Missel et al. [55], and Naarajan [56] t describe the transitin frm spherical t rdlike micelles. Fllwin the treatment presented by Israelachvili et al., startin frm the prpsed structure fr rdlike micelles shwn in Fiure.7 with a middle cylindrical part and quasi-spherical endcaps, ne can identify tw characteristic equilibrium cnstants assciated with the mlecules in the cylindrical part and thse in the endcaps, respectively. The standard chemical ptential f a rdlike micelle f size with cap mlecules in the tw spherical endcaps and ( cap ) mlecules in the cylindrical middle can be written as µ = ( ) µ + µ (.46) cap where µ cyl and µ cap are the standard chemical ptentials f the mlecules in the tw reins f the rdlike areate, respectively. Intrducin the abve relatin in the areate size distributin (Equatin.39), we btain cyl cap cap X µ cyl µ cap µ = X exp kt exp cap kt cyl (.47) where µ cyl and µ cap are the differences in the standard chemical ptentials between a surfactant mlecule in the cylindrical middle r the endcaps f the rdlike micelle and a sinly dispersed surfactant mlecule. Equatin.47 can be rewritten as X K Y Y X µ =, = exp cyl K kt, = exp cap µ µ cap kt cyl (.48) where K is a measure f the free enery penalty fr the mlecules present in the spherical endcap cmpared with thse in the cylindrical prtin. The averae areatin numbers can be cmputed frm Equatin.40 by analytically summin the series functins:

28 One Hundred Years f Micelles 9 Y Y = + Y = +, Y + Y + cap( Y ). (.49) n cap w cap Equatin.49 shws that fr values f Y clse t unity, very lare areates are frmed. The ttal cncentratin f surfactant present in the areated state is iven by the expressin: Y X = cap cap K Y + cap Y X ( Y) = T X. (.50) Here, X tt and X refer t the ttal amunt f surfactant and the amunt f surfactant present as sinly dispersed mlecules (practically, the cmc), s that the difference between them is the amunt f surfactant in the micellar frm. In the limit f Y clse t unity and cap ( Y), Equatin.50 reduces t X = X X K Y = T. (.5) Intrducin Equatin.5 in Equatin.49, the dependence f the averae areatin numbers n the surfactant cncentratin is btained. = + = + [ K( X X)] ( Y) n cap cap T w = cap + ( Y) = cap + [ K( XT X)] (.5) Because a realistic value fr X tt is less than 0 (that is, a surfactant cncentratin f 0.55 M r less), it is evident that K must be in the rane f 0 8 t 0 if lare rdlike micelles ( ~ 0 3 t 0 5 ) are t frm at physically realistic surfactant cncentratins. The plydispersity index ( w / n ) es frm unity t tw as the micelles rw frm lbules at lw surfactant cncentratin t iant rdlike micelles at hih surfactant cncentratins. These results are identical t the cnclusins reached by Mukerjee. These eneral thermdynamic results have since been used as the framewrk t evaluate the cnstant K, knwn in the literature as the sphere-t-rd transitin parameter, based n either experimental measurements f areate rwth with cncentratin [57,58] r by theretical mdelin f the free enery functins discussed in the fllwin sectins..9 tanford S PRINCIPLE OF OPPOSING FORCes Tanfrd prpsed the cncept f ppsin frces t frmulate a quantitative expressin fr the standard free enery chane n areatin, incrpratin quantitatively accurate expressins fr the hydrphbic interactins [8 30]. Tanfrd prpsed that the standard free enery chane assciated with the transfer f a surfactant frm its infinitely dilute state in water t an areate f size has three cntributins: µ µ µ kt = kt + kt + Tail Int µ kt. (.53) Head

29 30 Surfactant Science and Technly ( ) is a neative free enery cntributin arisin frm the transfer f the The first term µ / kt Tail tail frm its unfavrable cntact with water t the hydrcarbn-like envirnment f the areate cre. The transfer free enery cntributin depends n the surfactant tail but nt n the areate shape r size. The secnd term ( µ / kt ) prvides a psitive cntributin t accunt fr the fact Int that the entire surface area f the tail is nt remved frm water but there is still residual cntact with water at the surface f the areate cre. This is represented as the prduct f a cntact free enery per unit area σ (r an interfacial free enery) and the surface area per mlecule f the areate cre, a. The third term ( µ / kt ) prvides anther psitive cntributin representin Head the repulsive interactins between the head rups that crwd at the areate surface. The repulsins may be due t steric interactins (fr all types f head rups) and als electrstatic interactins (diple diple interactins fr zwitterinic head rups and in in repulsins fr inic head rups). Because the repulsin wuld increase if the head rups came clse t ne anther, Tanfrd prpsed an expressin with an inverse dependence n a. Thus, the standard free enery chane per mlecule n areatin prpsed by Tanfrd has the frm: µ µ σ α kt = kt + kt a + kt a Tail (.54) where α is the head rup repulsin parameter, k the Bltzmann cnstant, and T the temperature. Startin frm the free enery mdel f Tanfrd, the equilibrium areatin behavir can be examined either by treatin the surfactant slutin as cnsistin f areates with a distributin f sizes r by treatin the areate as cnstitutin a pseudphase. If the areate is viewed as a pseudphase, in the sense f small systems thermdynamics, the equilibrium cnditin crrespnds t a minimum in the standard free enery chane per mlecule, µ / kt. The minimizatin can be dne with respect t either the areatin number r the area per mlecule a because they are dependent n ne anther thruh the emetrical relatins iven in Table.. One btains in this manner, the equilibrium cnditin: µ = σ α 0 = 0 a kt kt kt a at e a = a = α σ / (.55) TABLE. Gemetrical Relatins fr Spherical and Cylindrical Micelles and Bilayers Variable Sphere Cylinder Bilayer Vlume f cre V = v 4πR 3 /3 πr R Surface area f cre A = a 4πR πr Area per mlecule a 3v /R v /R v /R Shape parameter P = v /ar /3 / Larest areatin number max 4π / 3v 4π /v l /v Areatin number max (3v /al ) 3 max (v /al ) max (v /al ) Nte: Variables V, A,, and max refer t the entire spherical areate, unit lenth f a cylinder, r unit area f a bilayer. R is the radius f spherical r cylindrical micelle r the half-bilayer thickness. v and l are the vlume and extended lenth f the surfactant tail. max is the larest areatin number pssible fr the iven emetry based n the cnstraint that the areate cre is filled and the tail cannt stretch beynd its extended lenth.

30 One Hundred Years f Micelles 3 The cmc (dented as X C in mle fractin units), in the pseudphase apprximatin, is btained frm the relatin, ln X C = kt kt a e µ + σ + α kt. (.56) a Tail In Tanfrd s free enery expressin (Equatin.54), the first cntributin, the tail transfer free enery, is neative (Fiure.9). Hence, this cntributin is respnsible fr the areatin t ccur. It affects nly the cmc (as shwn by Equatin.56) but nt the equilibrium area a e (as shwn by Equatin.55) r the size and shape f the areate. The secnd cntributin, the free enery f residual cntact between the areate cre and water, is psitive and decreases in manitude as the area a decreases. A decrease in the area a crrespnds t an increase in the areatin number, fr all areate shapes, as shwn in Table.. Hence, this cntributin prmtes the rwth f the areate. The third cntributin, the free enery due t head rup repulsins, is als psitive and increases in manitude if the area a decreases r the areatin number increases. Hence, this cntributin is respnsible fr limitin the rwth f areates t a finite size. Thus, Tanfrd s mdel clearly identifies why areates frm, why they rw, and why they d nt keep rwin but remain finite in size. Tanfrd s free enery expressin is widely recnized in the micellar literature and is cmmnly referred t as the principle f ppsin frces. Mst theretical wrk n micelles since 973 has been very much influenced by Tanfrd s frmalism. Indeed, we had used Tanfrd s mdel t prpse a theretical definitin fr the cmc as a transitin pint in the micelle size distributin [59,60], in cntrast with the usual practical definitin based n an bserved chane in the trend f any measured physical prperty as a functin f the surfactant cncentratin. Als, we used the Tanfrd mdel t predict that duble chain surfactant mlecules frm micelles as well as vesicles in slutins [34]. In frmulatin the free enery mdel, Tanfrd recnized that the state f the hydrcarbn tail f the surfactant within an areate is different frm that f a similar hydrcarbn in a bulk liquid state because ne end f the surfactant tail is cnstrained t remain at the areate water interface whereas the entire tail has t be packed within a iven areate shape maintainin liquid-like density in the areate cre. He accunted fr this empirically by makin a crrectin t the estimate fr the transfer free enery cntributin. Because this crrectin was taken t be independent f the areate shape and size, it had n effect n the size and shape f the areates. e 0 Free enery per mlecule (in kt) Transfer Surface Head rup Ttal Areatin number FIGURE.9 Cntributins t the free enery f micelle frmatin based n Tanfrd s principle f ppsin frces.

31 3 Surfactant Science and Technly Furthermre, in representin the surface free enery cntributin, Tanfrd did nt use the value f the macrscpic hydrcarbn water interfacial tensin fr the parameter σ appearin in Equatin.54. Instead, he estimated σ as equal t the manitude f the free enery chane per unit area in transferrin a hydrcarbn chain frm liquid hydrcarbn t water. The sum f the transfer free enery and the surface free enery terms in Tanfrd s mdel is exactly equivalent t the hydrphbic effect calculated by Pland and Scheraa. Finally, the head rup repulsins were represented by a phenmenlical expressin with an inverse dependence n the area per mlecule. A firm theretical basis t cmpute this cntributin fr nninic surfactants was nt available, althuh the phenmenlical expressin prvided nly an apprximatin fr electrstatic interactin eneries fr inic and zwitterinic head rups. All these features have been imprved upn in the recent predictive theries f micellizatin discussed in Sectin...0 MOLECULAR PACKING MODEL FOR SELF-ASSEMBLY Israelachvili et al. [3] prpsed the cncept f mlecular packin parameter and demnstrated hw the size and the shape f the areate at equilibrium can be predicted frm mlecular packin cnsideratins. The mlecular packin parameter is defined as v /al, where v and l are the vlume and the lenth f the surfactant tail and a is the surface area f the hydrphbic cre f the areate expressed per mlecule in the areate (usually referred t as the area per mlecule). If we cnsider a spherical micelle with a cre radius R, made up f mlecules, then the vlume f the cre V = v = 4πR 3 /3, the surface area f the cre A = a = 4πR, and hence R = 3 v /a, frm simple emetrical relatins (Table.). If the micelle cre is packed with surfactant tails withut any empty space, then the radius R cannt exceed the extended lenth l f the tail. Intrducin this cnstraint in the expressin fr R, ne btains the cnstraint n the mlecular packin parameter, that is, 0 v /al /3, fr spherical micelles. Fr spherical, cylindrical, r bilayer areates, the emetrical relatins fr the vlume V and the surface area A are iven in Table.. The variables V, A, and in the table refer t the entire spherical areate, unit lenth f a cylindrical areate, r unit area f a bilayer areate, respectively, fr the three shapes. These emetric relatins, tether with the cnstraint that at least ne dimensin f the areate (the radius f the sphere r the cylinder, r the half-bilayer thickness, all dented by R) cannt exceed l, lead t the fllwin well-knwn [3] cnnectin between the mlecular packin parameter and the areate shape: 0 v /al /3 fr spheres, /3 v /al / fr cylinders, and / v /al fr bilayers. Therefre, if we knw the mlecular packin parameter, the shape and size f the equilibrium areate can be readily identified, as shwn abve. This is the predictive sense in which the mlecular packin parameter f Israelachvili et al. [3] has fund sinificant use in the literature. The ntin f mlecular packin int varius areate shapes was recnized even in the earlier wrks f Tartar [6] and Tanfrd [8], as can be seen, fr example, frm Fiure 9. f Tanfrd s classic mnraph. Hwever, nly after this cncept was explred thruhly in the wrk f Israelachvili et al., takin the frm f the packin parameter, did it evke wide appreciatin in the literature. Fr cmmn surfactants, the rati v /l is a cnstant independent f tail lenth, equal t Ǻ fr a sinle tail and 4 Ǻ fr a duble tail [8]. Cnsequently, nly the area a e reflects the specificity f the surfactant in the packin parameter v /a e l. The area per mlecule a e is a thermdynamic quantity btained frm equilibrium cnsideratins f minimum free enery and is nt a simple variable cnnected t the emetrical shape and size f the surfactant head rup. Israelachvili et al. invked the free enery mdel f Tanfrd [8] t estimate the equilibrium area per mlecule, a e. µ µ σ α kt = kt + kt a + µ kt = = α, 0 at ae (.57) a a kt σ Tail /

32 One Hundred Years f Micelles 33 The packin parameter v /a e l is readily calculated usin the area a e btained frm Equatin.57. One can bserve that a e will be small and the packin parameter will be lare if the head rup interactin parameter α is small. The area a e will increase and the packin parameter will decrease if the interfacial free enery per unit area σ decreases. These simple cnsideratins allw ne t predict many features f surfactant self-assembly as summarized belw..0. nninic Surfactants Fr nninic surfactants with liethylene xide as the head rup, the steric repulsins between the head rups can be expected t increase if the number f ethylene xide units in a head rup increases. Crrespndinly, the head rup parameter α will increase in manitude if the number f ethylene xide units in the head rup increases. Therefre, when the number f ethylene xide units is small, α is small, a e is small, v /a e l is lare, and bilayer areates (lamellae) are favred. Fr a larer number f ethylene xide units, α increases, a e increases, v /a e l decreases, and cylindrical micelles becme pssible. When the number f ethylene xide units is further increased, v /a e l becmes small enuh s that spherical micelles will frm, with their areatin number decreasin with increasin ethylene xide chain lenth. Furthermre, the increase in α and a e fr the nninic surfactants with increasin ethylene xide chain lenth ives rise t an increase in the cmc as the head rup size increases. Fr a iven ethylene xide chain lenth and an equilibrium packin parameter crrespndin t spherical micelles, increasin the chain lenth f the surfactant tail will cause an increase in the areatin number f the micelles as predicted by the emetrical relatin in Table...0. inic and Zwitterinic Surfactants Cmparin inic and nninic surfactants, the head rup interactin parameter α will be larer fr the inic surfactants than fr the nninic surfactants because ne has t als cnsider inic repulsins between the head rups in the frmer case. Therefre, the equilibrium area per mlecule, a e will be larer and v /a e l will be smaller fr the inics cmpared with the nninics. As a result, inic surfactants wuld frm areates f smaller areatin number cmpared with nninic surfactants f the same tail lenth. Furthermre, the increase in α and a e fr the inic surfactants als ives rise t an increase in the cmc when inic and nninic surfactants f equal tail lenths are cmpared. The head rup repulsins fr zwitterinic surfactants are intermediate between thse fr inic and nninic surfactants. Therefre, bth the cmc fr zwitterinic surfactants and their areatin numbers will be intermediate between thse fr inic and nninic surfactants f the same tail lenth..0.3 Slutin Cnditins Fr a iven surfactant mlecule, the head rup repulsin can be mdified by a chane in the slutin cnditins. Fr example, addin salt t an inic surfactant slutin decreases electrstatic repulsins between inic head rups; increasin the temperature fr a nninic surfactant mlecule with ethylene xide head rup decreases steric repulsins between the nninic head rups. Because the head rup repulsin parameter α decreases, the equilibrium area per mlecule, a e, will decrease and the packin parameter v /a e l will increase. Thus, ne can achieve a transitin frm spherical micelles t rdlike micelles and pssibly t bilayer areates, by mdifyin slutin cnditins (addin salt t inic surfactants and increasin the temperature fr the nninic surfactants) that cntrl head rup repulsins..0.4 duble Tail Surfactants If sinle tail and duble tail surfactant mlecules are cmpared, fr the same equilibrium area per mlecule, a e, the duble tail mlecule will have a packin parameter v /a e l twice as lare as that

33 34 Surfactant Science and Technly f the sinle tail mlecule. Therefre, the duble tail mlecule can self-assemble t frm bilayer vesicles whereas the crrespndin sinle tail mlecule areates int nly spherical r lbular micelles..0.5 Plar Oranic Slvents If the slvent is chaned frm water t a mixed aqueus ranic slvent, then the interfacial tensin parameter σ decreases. Fr a iven surfactant, this wuld lead t an increase in the equilibrium area per mlecule a e, and hence, a decrease in v /a e l. Therefre, upn the additin f a plar ranic slvent t an aqueus surfactant slutin, bilayers will transfrm int micelles, rdlike micelles int spherical micelles, and spherical micelles int thse f smaller areatin numbers includin nly small mlecular clusters. Startin frm phsphlipid surfactants that have tw hydrphbic tails per mlecule, we can frm bilayer vesicles, cylindrical micelles, r spherical micelles, respectively, if we use an aqueus ranic slvent mixture with increasin amunts f plar ranic slvent in the slvent mixture. All f the abve predictins are in areement with numerus experiments and are by nw well established in the literature (see numerus experimental studies that have been included in the Surfactants in Slutin series f bks as well as in varius jurnals). One can thus see evidence f the predictive pwer f the phenmenlical free enery mdel and the mlecular packin cnsideratins, despite their remarkable simplicity. These cncepts can be used t predict the selfassembly behavir f ther nvel amphiphiles and ne example is the predictins made fr bla amphiphiles that have a sinle hydrphbic chain separatin tw terminal plar head rups in each mlecule [6]. The cncept f a mlecular packin parameter has been widely cited in the literature fr chemistry, physics, and bily because it allws a simple and intuitive insiht int the selfassembly phenmenn [63 65]. Withut any dubt, the cntributins f Tanfrd and f Israelachvili et al. have had a lastin effect n the surfactant literature ver the past 35 years. In the packin parameter mdel utilizin the free enery expressin f Tanfrd, the surfactant tail has n recnizable rle in determinin the shape and size f the self-assembled structure. We have prpsed a mdified free enery mdel that takes int accunt tail-packin cnstraints inside the micelle cre [66]. The inclusin f this cntributin intrduces an explicit as well as an implicit dependence f the areatin prperties n the tail lenth, in additin t the accepted dependence n the head rup. We have shwn that the tail lenth s effects are critical, especially fr predictin when the transitins between areate shapes ccur [67]. We revisit the packin parameter mdel in Sectin., takin advantae f the mre refined free enery expressins we have develped as part f a predictive thery f surfactant self-assembly.. MOLECULAR SCALE PREDICTIVE THeRIes OF MICELLE FORMATIn Many f the eneral principles f surfactant areatin have nw been adequately clarified by the theretical studies mentined. Yet, these studies have nt prvided quantitative a priri predictins f areatin behavir startin frm the mlecular structure f the surfactants and the slutin cnditins such as temperature, inic strenth, surfactant cncentratin, etc. Fr example, in the free enery mdel f Tanfrd, sme cntributins were estimated by utilizin experimental data available fr micellar slutins. First, in writin the transfer free enery term, Tanfrd started with an expressin fr the transfer f a hydrcarbn chain frm bulk water t a bulk hydrcarbn liquid. Then, recnizin that the tail cnfrmatin in the micelle interir is different frm that in a bulk hydrcarbn liquid, he prpsed an empirical crrectin t the free enery t ensure areement with the experimental cmc data. Despite its empirical nature, this crrectin term had universality in the sense that it was nt altered frm ne surfactant t anther but was cnsidered valid fr all surfactants. This empirical expressin assumed the chain cnfrmatin enery t have n influence n the areate size and shape. Secnd, the free enery cntributin due t the

34 One Hundred Years f Micelles 35 electrstatic interactins between the plar head rups f inic surfactants was estimated usin the Debye Hückel apprximatin in which an empirical crrectin was incrprated. A cnstant numerical factr f abut half (0.46 bein the best fit) in the Debye Hückel expressin was bserved t be necessary t describe the experimental cmc data. Aain, this empirical cnstant was nt chaned fr different inic surfactants but was used as a universal crrectin cnstant. Third, n explicit expressin dependent n the head rup was bvius in the Tanfrd mdel t describe head rup interactins in nninic surfactants. Furthermre, fr the nninic surfactants cntainin ply(xyethylene) chain plar head rups, the interactins amn the head rups may require a mdel different frm ne that culd be used fr the mre cmpact head rups such as lucsides. Finally, Tanfrd treated the surface enery usin estimates f hydrphbic bnd enery per unit area rather than usin the well-knwn hydrcarbn water interfacial tensin. The free enery mdels develped in ur wrk [66] and by the Blankschtein rup [68] have attempted t eliminate these empirical features t allw fr truly a priri predictins f the areatin prperties, iven the surfactant mlecular structure. We describe nly ur wrk in detail belw but wuld direct the readers t explre many imprtant articles frm the Blankschtein rup [69 76], whse apprach is bradly similar t urs but differin in many imprtant details. In cnstructin the free enery mdel, we d nt utilize any infrmatin derived frm experiments n surfactant slutins t ensure that the predictins frm the mdel remain truly a priri. T arrive at such a free enery expressin, first we develped an analytical equatin fr the chain cnfrmatin free enery that is dependent n the size and the shape f the areates by brrwin results frm the analysis f the chain cnfrmatin in cplymer melts [77]. This cntributin allwed new predictins t be made that were verlked by the empirical, shape-independent free enery crrectin term used earlier by Tanfrd and als in ur earlier mdelin [78 8]. Secnd, the electrstatic interactins in inic micelles were cmputed usin an apprximate analytical slutin t the Pissn Bltzmann equatin derived by Evans and Ninham [8], which incrprates curvature crrectins. The use f this equatin is fund t prvide a satisfactry estimate f the inic interactin free enery and avids the earlier use f the empirical cefficient f 0.46 in cnjunctin with the Debye Hückel apprximatin. Third, the interactins amn ply(xyethylene) head rups in nninic micelles was treated by takin int accunt the plymeric nature f the plar head rup and cnsiderin the mixin and defrmatin free eneries in the head rup rein in additin t the steric repulsins at the micellar cre surface. Furth, the surface enery is treated usin macrscpic il water interfacial tensin. Finally, the temperature dependences f all the free enery cntributins are explicitly described, allwin ne t calculate the temperature dependence f the areatin behavir. The details f the predictive thery are briefly presented belw. Mre extensive discussins includin cmparisns with experimental data can be fund in ur previus publicatins [66,83 86]. We emphasize aain that in develpin the free enery mdel, we have nt made use f any infrmatin btained a psteriri frm experimental areatin data. Obviusly, varius mlecular cnstants are invlved in the calculatins but they can be readily estimated frm the mlecular structural prperties f surfactants... cntributins t Free Enery Chane n Areatin ( ) between a surfactant mlecule in In ur mdel [66], the standard free enery difference µ / kt an areate f size and ne in the sinly dispersed state is decmpsed int a number f cntributins n the basis f mlecular cnsideratins. First, the hydrphbic tail f the surfactant is remved frm cntact with water and transferred t the areate cre, which is like a hydrcarbn liquid. Secnd, the surfactant tail inside the areate cre is subjected t packin cnstraints because f the requirements that the plar head rup shuld remain at the areate water interface and the micelle cre shuld have a hydrcarbn liquid like density. Third, the frmatin f the areate is assciated with the creatin f an interface between its hydrphbic dmain and water.

35 36 Surfactant Science and Technly Furth, the surfactant head rups are bruht t the areate surface ivin rise t steric repulsins between them. Last, if the head rups are inic r zwitterinic, then electrstatic repulsins between the head rups at the areate surface als arise. Explicit analytical expressins fr each f these free enery cntributins were develped [66] in terms f the mlecular characteristics f the surfactant and they are briefly discussed in this sectin... transfer f the Surfactant Tail The cntributin t the free enery frm the transfer f surfactant tail frm water t the micelle cre is estimated by cnsiderin the cre t be similar t a liquid hydrcarbn. This allws us t estimate the transfer free enery usin cmpiled experimental data n the slubility f hydrcarbns in water [87,88]. Nte that n infrmatin derived frm experiments invlvin surfactants is used anywhere in ur mdel. We have estimated [66] the rup cntributins t the transfer free enery fr a methylene and a methyl rup in an aliphatic tail t be µ 896 T T fr t kt = ln ( he CH rup ) T Tr µ ln T. kt = T ( fr the CH3 rup). T Tr (.58) Because the micelle cre differs frm a liquid hydrcarbn, a cntributin t the free enery frm this different state shuld be cnsidered...3 defrmatin f the Surfactant Tail In frmulatin his phenmenlical free enery mdel, Tanfrd recnized that the state f the hydrcarbn tail f the surfactant within an areate is different frm that f a similar hydrcarbn in bulk liquid state because ne end f the surfactant tail is cnstrained t remain at the areate water interface whereas the entire tail has t be packed within a iven areate shape maintainin liquid-like density in the areate cre (Fiure.0). He accunted fr this empirically by makin a crrectin t the estimate fr the transfer free enery cntributin. Because this crrectin was taken t be independent f the areate shape and size, it culd nt describe any effect f the mlecular packin cnstraints n the size and shape f the areates. Detailed chain-packin mdels t estimate this free enery cntributin ( µ / kt ) as a functin f the areate shape and size have been develped fllwin different appraches by Gruen def [9 93], Dill and Flry [90,94 96], and Ben-Shaul et al. [97 99]. In ur wrk, we have adapted the methd suested fr blck cplymers by Semenv [77] t btain an analytical expressin fr the tail defrmatin enery [66]. Fr spherical micelles, µ Pπ R kt = 9 NL (.59) 80 Def where P is the shape parameter defined in Table., R is the cre radius, L is the sement lenth and, N is the number f sements in the tail (N = l /L, where l is the extended lenth f the tail). As suested by Dill and Flry [90,94], a sement is assumed t cnsist f 3.6 methylene rups (hence, L = 0.46 nm). L als represents the spacin between alkane mlecules in the liquid state, namely, L = 0. nm is the crss-sectinal area f the plymethylene chain. Equatin.59 is als used fr lbular micelles and the spherical endcaps f rdlike micelles. Fr the cylindrical middle

36 One Hundred Years f Micelles 37 nm FIGURE.0 In the micelle, the methylene rup f the surfactant tail that is attached t the plar head rup is cnstrained t remain at the areate water interface. The ther end (the terminal methyl rup) is free t ccupy any psitin inside the areate as ln as unifrm liquidlike density is maintained in the areate cre. In the classic picture f a Hartley micelle, these packin features are nt recnized because the radial rientatin f the tails is incnsistent with the requirement f unifrm liquidlike density in the cre. Gruen [89] presented the space-fillin mdel fr a micelle cnsistin f apprximately 55 mlecules with ddecyl hydrphbic chain. Clearly, it is necessary fr the tail t defrm nnunifrmly aln its lenth t satisfy bth the packin and the unifrm density cnstraints. Dill and Flry [90] develped a lattice representatin fr the packin inside the micelle meetin the liquidlike density requirement in the cre. (Left and riht: Reprinted frm K.A. Dill and P.J. Flry, Prc. Natl. Acad. Sci. USA, 78, , 98. With permissin. Middle: With kind permissin frm Sprin Science + Business Media: Prr. Cllid Plym. Sci., 70, 985, 6 6, D.W.R. Gruen.) f rdlike micelles, the cefficient 9 is replaced by 0, the radius R nw represents the radius f the cylinder, and P = /. Fr bilayers, the cefficient 9 is replaced by 0, the radius R represents the half-bilayer thickness, R and P =...4 frmatin f Areate Cre Water Interface Micelle frmatin enerates an interface between the hydrphbic cre and the surrundin water medium. The free enery f frmatin f this interface is calculated as the prduct f the surface area in cntact with water and the macrscpic interfacial tensin σ a characteristic f the interface [66]: µ σ a a kt = a kt ( ). (.60) int Here, a is the surface area f the micelle cre per surfactant mlecule, and a is the surface area per mlecule shielded frm cntact with water by the plar head rup f the surfactant. Expressins fr the area per mlecule a crrespndin t different areate shapes are prvided in Table.. The area a depends n the extent t which the plar head rup shields the crsssectinal area L f the surfactant tail. If the head rup crss-sectinal area a p is larer than L, the tail crss-sectin is shielded cmpletely frm cntact with water and a = L. If a p is smaller than L, then the head rup shields nly a part f the crss-sectinal area f the tail frm cntact with water, and in this case, a = a p. Thus, a is equal t the smaller f a p and L. The areate cre water interfacial tensin σ a is taken equal t the interfacial tensin σ SW between water (W) and the aliphatic hydrcarbn f the same mlecular weiht as the surfactant tail (S). The interfacial tensin σ SW can be calculated in terms f the surface tensins σ S f the aliphatic surfactant tail and σ W f water [00]: