1 Molecular. Thermodynamics of Giant Micelles. Ramanathan Nagarajan CONTENTS

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3 1 Mlecular Thermdynamics f Giant Micelles Ramanathan Naarajan CONTENTS 1.1 Intrductin Thermdynamic Principles f Giant Micelles Gemetry f Giant Micelles Slutin Mdel f Micelle Frmatin Frmatin f Rdlike Micelles Free Enery Chane n Giant 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 Head Grup Interactins Diplar Head Grup Interactins Inic Mdel Predictins f Giant Micelles Cmputatinal Apprach Estimatin f Mlecular Cnstants Estimatin f Tail Vlume v S Estimatin f Extended Tail Lenth l S Estimatin f Head Grup Area a p Estimatin f δ and d Illustrative Predicted Results Tunin Structural Transitins in Giant Micelles Surfactant Mixtures Free Enery f Frmatin f Mixed Micelles Estimatin f Mlecular Cnstants and Cmputatinal Apprach Predicted Results...6 1

4 Giant Micelles: Prperties and Applicatins 1.5. Additin f an Alchl Additin f Nninic Plymers Free Enery f Micellizatin in the Presence f Plymer Plymer Induced Sphere t Rd Transitin Cnclusins...37 Acknwledments...38 References INTRODUCTION The first mlecular thery f micelle frmatin was prpsed by Debye. 1, Fr the lamellar micelle suested by Mcain, 3,4 Debye frmulated tw cmpetin eneries cntrllin micelle frmatin: the repulsive interactins between inic head rups n the micelle surface and the attractive van der Waals interactins between the tails. y minimizin the enery f a micelle, its equilibrium size and the critical micelle cncentratin (cmc) were btained. The thery was extended by Hbbs 5 t accunt fr electrlyte effects. Halsey 6 amended the enery expressin t cnsider cylinders and cncluded that ne-dimensinal areate rwth will be as rds and nt as disks. Oshika 7 and Reich 8 added a surface enery cntributin t micelle frmatin. They als arued that the enery f the entire system, and nt the enery f a micelle, shuld be minimized. Later, partitin functins based n statistical thermdynamics were develped by Heve and ensn 9 and Pland and Scheraa. 10,11 Subsequently, Tanfrd 1 14 prpsed his influential principle f ppsin frces with three free enery terms: (i) a neative cntributin due t remval f tail-water cntact, which is respnsible fr micelle frmatin, (ii) a psitive cntributin due t residual tail-water cntact which prmtes micelle rwth, and (iii) a psitive cntributin due t head rup repulsins which limits micelles t finite sizes. Takin advantae f these earlier theretical studies, ur wrk has fcused n develpin a priri predictive mdels f areatin behavir f surfactants startin frm knwlede f their mlecular structures and the envirnmental cnditins in which they are present. Our first attempt 15 invlved statistical thermdynamic frmulatins influenced by the wrk f Heve and ensn 9 and Pland and Scheraa 10,11 and led t the develpment f partitin functins t represent varius cntributins t the free enery f micellizatin. Subsequently, we recast 16 0 the partitin functins in the frm f free enery cntributins similar t thse emplyed in Tanfrd s frmalism. In ur apprach, the physicchemical factrs cntrllin self-assembly are first identified by examinin all the chanes experienced by a sinly dispersed surfactant mlecule when it becmes part f an areate. T calculate the cntributin t the free enery f areatin assciated with each f these factrs, analytical equatins explicitly cnnected t the mlecular features f the surfactant are frmulated. The chemical structure f the surfactant and the slutin cnditins are sufficient fr estimatin all the mlecular cnstants appearin in these equatins. The main advantae f ur theretical mdel lies in the analytical

5 Mlecular Thermdynamics f Giant Micelles 3 frm f free enery expressins, the need t estimate nly ne, tw, r three mdel parameters dependin upn the type f surfactant (see Sectin 1.4. belw), and the ability t make a priri predictins requirin simple numerical calculatins. Further, a wide rane f self-assembly phenmena such as slubilizatin, 17,0 mixed micelles, 18,19,1 and micremulsins are treated by simple extensins f the basic micellizatin thery allwin fr a unified apprach t be develped. In this chapter, we describe ur quantitative apprach t predictin the frmatin and prperties f rdlike micelles and the shape transitins they under. In Sectin 1., we summarize the emetrical relatins fr spherical, lbular, and rdlike micelles, cnsistent with mlecular packin cnsideratins, and develp eneral thermdynamic equatins that vern the frmatin f rdlike micelles. In Sectin 1.3, the mlecular thery f rdlike micelle frmatin is described, with explicit equatins t calculate the free enery f frmatin f areates. In Sectin 1.4, we shw hw t estimate the mlecular cnstants appearin in the free enery expressins, describe the cmputatinal scheme fr makin predictive calculatins, and demnstrate the predictive pwer f the mlecular thery via illustrative calculatins. In Sectin 1.5, we cnsider hw rdlike micelles can be manipulated in varius ways such as by the additin f a secnd surfactant, by the additin f an alchl, and by the additin f a nninic plymer. Fr this purpse, we present extensins f the mlecular thery described in Sectin 1.3 t the prblems f mixed micelles, and surfactant plymer assciatin. In each case, illustrative calculatins are presented t shw hw rdlike micelles are manipulated. The last sectin presents sme cnclusins. 1. THERMODYNAMIC PRINCIPLES OF GIANT MICELLES 1..1 GEOMETRY OF GIANT MICELLES The shapes f surfactant areates frmed in dilute slutins are schematically shwn in Fiure 1.1. The small micelles are spherical. When lare rdlike micelles frm, they are visualized as havin a cylindrical middle part and spherical endcaps, with the tw parts allwed t have different diameters. Fr cnditins when spherical micelles are nt pssible (namely fr thse areatin numbers crrespndin t the sphere that will have a radius larer than the extended tail lenth l s ) and at the same time, the frmatin f rdlike micelles is als nt favred, small nnspherical lbular areates frm. Israelachvili et al. 3 evaluated varius shapes based n the cnstraint that mlecular packin requirements shuld be satisfied lcally everywhere n the areate. On this basis, they have suested shapes enerated by ellipses f revlutin fr these lbular areates. The averae area per surfactant mlecule at the surface f the ellipse f revlutin is practically the same as that f prlate ellipsids, fr areatin numbers up t three times that f the larest spherical micelle. Therefre, we will calculate the averae emetrical prperties f lbular areates in the sphere-t-rd transitin rein as fr prlate ellipsids (withut implyin the shape t be an

6 4 Giant Micelles: Prperties and Applicatins Spherical micelle Glbular micelle Rdlike micelle Spherical bilayer vesicle FIGURE 1.1 Schematic representatin f surfactant areates in dilute aqueus slutins. 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. ellipsid). Sme surfactants pack int a spherical bilayer structure called a vesicle, which enclses an aqueus cavity. We will nt cnsider vesicles in this chapter even thuh a transitin between vesicles and rdlike micelles is pssible and has been predicted by ur thery. Fr spherical, lbular, and cylindrical areates cntainin surfactant mlecules, the vlume f the hydrphbic cre f the areate, V, the surface area f cntact between the areate cre and water, A, and the surface area f the areate at a distance δ frm the areate cre water interface, A δ, are listed in Table 1.1. Als iven in the table is a packin parameter p, defined in terms f the emetrical variables characterizin the areate. The packin parameter p is used in the cmputatin f the free enery f tail defrmatin (Sectin 1.3.3) and the area A δ is emplyed in the cmputatin f the free enery f electrstatic interactins between surfactant head rups (Sectins and 1.3.7). Given a surfactant mlecule, the emetrical prperties f spherical r lbular micelles depend nly n the areatin number. Fr rdlike micelles, there are tw independent variables, the radius f the cylindrical part and the radius f the spherical endcaps.

7 Mlecular Thermdynamics f Giant Micelles 5 TALE 1.1 Gemetrical Prperties f Surfactant Micelles 3 4 π RS V = = v, A = 4 π R = a S S 3 A = 4 π( R + δ) = a δ S S V vs 1 p = = = AR ar 3 S Spherical micelles: (Radius R S l S ) δ Glbular micelles: (Semiminr axis R S = l S, Semimajr axis b 3l S, Eccentricity E) 4 π R b 1 S sin E V = = v, A = π R 1+ S S 1 3 E ( 1 E ) / = = RS a, E 1 b 1 A = + + sin E δ π( R δ) 1 δ 1 = a, 1 E ( E ) / δ δ V p = A R S S δ δ v V S 1 3 =, p , R = eq ar 3 S 4 π / 1 R + δ S E = 1 b + δ Cylindrical part f rdlike micelles: (Radius R C l S, Lenth L C ) V = π R L = v, A = π R L = a C C S C C A = π( R + δ) L = a δ C C δ V vs p = = = 1 AR ar C C / 1 Endcaps f rdlike micelles: (Endcap radius R S l S, Cylinder radius R C l S ) V 3 8 π RS = π H R H v A R S = = ( 3 ), 8π 4 S S 3 3 π R H = a S = A 8 π( R + δ) 4 π( R + δ)( H+ δ) δ S S = a, H= R R R δ S C S ( / ) / V p = = v S AR ar S S 13 / { }

8 6 Giant Micelles: Prperties and Applicatins 1.. SOLUTION MODEL OF MICELLE FORMATION The surfactant slutin is a binary mixture f the surfactant and water. Hwever, we can view it as a multicmpnent system by treatin the areate f each size and shape as a distinct chemical cmpnent described by a characteristic chemical ptential. 1,4,5 We cnsider a slutin cnsistin f N W water mlecules, N 1 sinly dispersed surfactant mlecules, and N areates f all pssible shapes and areatin number, with allwed t take all values frm t. The subscript W refers t water, 1 t the sinly dispersed surfactant, and t the areate cntainin surfactant mlecules. The Gibbs free enery f the slutin G is the number-weihted sum f the chemical ptentials µ i f the varius species i. = G = N µ + N µ + N µ w w 1 1 = (1.1) The equilibrium cnditin f a minimum in the free enery leads t µ µ = 1 (1.) This equatin 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. Fr a dilute surfactant slutin, ne can write (fr all values f includin = 1) µ = µ + ktln X (1.3) µ where is the standard state chemical ptential f the specie, X is its mle fractin in slutin, k is the ltzmann cnstant, and T is the abslute temperature (expressed everywhere in this chapter in K). 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 abve tw equatins, we et the areate size distributin X µ µ 1 = X exp X kt k = exp µ 1 1 T (1.4) µ where is the difference in the standard chemical ptentials between a surfactant mlecule present in an areate f size and a sinly dispersed surfactant

9 Mlecular Thermdynamics f Giant Micelles 7 in water. T calculate the areate size distributin, we need an explicit equatin fr µ. The critical micelle cncentratin (cmc) can be calculated frm the areate size distributin 6,7 by cnstructin a plt f any ne f the functins X 1, ΣX, ΣX, r Σ X (which are prprtinal t different experimentally measured prperties f the surfactant slutin such as surface tensin, electrical cnductivity, dye slubilizatin, liht scatterin intensity, etc.) aainst the ttal surfactant cncentratin X tt (= X 1 + ΣX ) and equatin the cmc t the ttal surfactant cncentratin at which a sharp chane in the pltted functin ccurs. The cmc can als be estimated 7 as the value f X 1 fr which the amunt f surfactant in the micellized frm is equal t that in the sinly dispersed frm. Yet anther apprach 7 t estimate the cmc is t equate it t the ttal surfactant cncentratin fr which abut 5% f the ttal surfactant is in the frm f micelles. Frm the micelle size distributin we can cmpute averae areatin numbers usin the definitins n 3 ΣX Σ X Σ X =, =, = w z ΣX ΣX Σ X (1.5) 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. 5 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 8 frm the size distributin that the averae areatin numbers n and w depend n the cncentratin f the micellized surfactant as fllws: = = n z ln 1 ln X, ln 1 n w ln X w w ( 1 n/ w) ( z/ w 1) X, X n w (1.6) 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 averae areatin numbers must be virtually independent f the ttal surfactant cncentratin if the micelles are narrwly dispersed. Further, Equatin 1.6 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 mdels fr micellizatin.

10 8 Giant Micelles: Prperties and Applicatins 1..3 FORMATION OF RODLIKE MICELLES Rdlike micelles are visualized as havin a cylindrical middle part with tw spherical endcaps as shwn in Fiure 1.1. Mukerjee 9 33 was the first t treat the thermdynamics f rdlike areates by recnizin that tw characteristic equilibrium cnstants are necessary. Mukerjee used a dimerizatin cnstant and, fr areates larer than dimers, a cnstant stepwise assciatin equilibrium cnstant. Hwever, we can nw identify the tw relevant equilibrium cnstants with the mlecules in the cylindrical part and thse in the endcaps, respectively. The thermdynamic frmulatin fr rdlike micelles has been presented by many authrs 3,34 36 usin smewhat different terminlies, but the equivalence between all f these treatments and their cnnectin t Mukerjee s pineerin wrk has been discussed in ur previus wrk. 37 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 cap cyl cap cap (1.7) µ cyl µ cap where and 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 we et X cyl = X exp µ exp µ µ cap 1 cap kt kt cyl (1.8) 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 1.8 can be rewritten as X 1 K Y Y X µ cyl =, = exp, 1 kt K = exp cap µ µ cap kt cyl (1.9) where K is a measure f the free enery penalty fr the mlecules present in the spherical endcap cmpared t thse in the cylindrical prtin. The averae areatin numbers can be cmputed frm Equatin 1.9 by analytically summin the series functins. 0,3 Y Y 1 = + Y, = Y Y + ( 1 Y) cap n cap w cap (1.10)

11 Mlecular Thermdynamics f Giant Micelles 9 Equatin 1.10 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 0,3 X 1 = K cap Y cap 1 Y 1 + cap Y = X ( 1 Y) tt X 1 (1.11) Here, X tt and X 1 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 (1 Y) << 1, Equatin 1.11 reduces t X 1 1 = X X tt K 1 Y = 1 (1.1) Intrducin Equatin 1.1 in Equatin 1.10 yields the dependence f the averae areatin numbers n the surfactant cncentratin. 1 1 / = + = + [ K ( X X )], 1 1 Y n cap cap tt w = + = + [ K ( X X )] / cap cap tt 1 1 Y 1 (1.13) Since a realistic value fr X tt is less than 10 (that is, a surfactant cncentratin f 0.55 M r belw), it is evident that K must be in the rane f 10 8 t 10 1, if lare rdlike micelles ( ~ 10 3 t 10 5 ) are t frm at physically realistic surfactant cncentratins. The plydispersity index ( w / n ) es frm unity t as the micelles rw frm the smallest rds at lw surfactant cncentratin t iant rdlike micelles at hih surfactant cncentratins. The cmc is apprximately estimated frm Equatin 1.9 based n the cnditin that Y is clse t unity. X cmc = cyl exp µ kt (1.14) The thermdynamic results btained s far are independent f any specific expressin fr the standard free enery chane µ assciated with areatin and cnstitute eneral theretical principles vernin the areatin behavir f surfactants. Hwever, t perfrm predictive calculatins f the areatin behavir, specific expressins fr µ are needed and these are develped in the fllwin sectin.

12 10 Giant Micelles: Prperties and Applicatins 1.3 FREE ENERGY CHANGE ON GIANT MICELLE FORMATION CONTRIUTIONS TO FREE ENERGY CHANGE ON AGGREGATION The standard free enery difference µ between a surfactant mlecule in an areate f size and ne in the sinly dispersed state can be decmpsed int a number f cntributins n the basis f mlecular cnsideratins. 0 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 liquidlike density. Third, the frmatin f the areate is assciated with the creatin f an interface between its hydrphbic dmain and water. 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 have been develped in ur earlier studies 0 in terms f the mlecular characteristics f the surfactant, and they are briefly discussed in this sectin TRANSFER OF 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 like a liquid hydrcarbn. This allws us t estimate the transfer free enery usin cmpiled experimental data n the slubility f hydrcarbns in water. 38,39 Nte that n infrmatin derived frm experiments invlvin surfactants is used anywhere in ur mdel. We have estimated 0 the rup cntributins t the transfer free enery fr a methylene and a methyl rup in an aliphatic tail t be ( ) µ tr kt ( µ ) tr kt 896 = 5.85 ln T T (fr CH ) T 4064 = 3.38 ln T T (fr CH 3 ) T (1.15) Since the micelle cre differs frm a liquid hydrcarbn, a cntributin t the free enery frm this different state is cnsidered next.

13 Mlecular Thermdynamics f Giant Micelles DEFORMATION OF THE SURFACTANT TAIL In the surfactant tail, the methylene rup 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. It is necessary fr the tail t defrm nnunifrmly aln its lenth in rder t satisfy bth the packin and the unifrm density cnstraints. The psitive free enery cntributin resultin frm this cnfrmatinal cnstraint n the tail is referred t as the tail defrmatin free enery. y adaptin the methd suested fr blck cplymers by Semenv, 40 we have derived a simple analytical expressin fr the tail defrmatin enery. 0 Fr spherical micelles ( ) = µ def 9 kt π p R S 80 NL (1.16) where p is the packin factr defined in Table 1.1, R S is the cre radius, L is the sement lenth and N is the number f sements in the tail (N = l S /L, where l S is the extended lenth f the tail). As suested by Dill and Flry, 41,4 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 (= 0.1 nm ) is the crss-sectinal area f the plymethylene chain. Equatin 1.16 is used als fr lbular micelles and the spherical endcaps f rdlike micelles. Fr the cylindrical middle f rdlike micelles, the cefficient 9 is replaced by 10, the radius R S is replaced by the radius R C f the cylinder, and p = 1/ FORMATION OF AGGREGATE CORE 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 0 ( ) int = µ σ a ( a a ) kt kt (1.17) 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 1.1. The area a depends n the extent t which the plar head rup shields the crss-sectinal area L f the surfactant tail. If the head rup crss-sectinal area a p is larer than L, the

14 1 Giant Micelles: Prperties and Applicatins 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 43 σ = σ + σ.0 ψ( σ σ ) 1/ SW S W S W 3 / σ = M ( T 98) S σ W = ( T 98) (1.18) In Equatin 1.18, ψ is a cnstant 43 equal t 0.55, M is the mlecular weiht f the surfactant tail, and the expressins fr σ S and σ W (in mn/m) are based n literature data. 0,43, HEAD GROUP INTERACTIONS STERIC Upn micelle frmatin, the surfactant head rups are bruht t the micelle surface where they are crwded when cmpared t the infinitely dilute state f the sinly dispersed surfactant mlecules. This enerates steric repulsins amn the head rups. Fr cmpact head rups, by analy with the repulsin term in the van der Waals equatin f state, we have prpsed 15,16,0 the expressin ( µ ) steric = ln 1 kt a p a (1.19) where a p is the crss-sectinal area f the plar head rup. This equatin is used fr spherical and lbular micelles as well as fr the cylindrical middle and the spherical endcaps f rdlike micelles. This apprach t calculatin the steric interactins is inadequate when the plar head rups are nt cmpact, as in the case f nninic surfactants with liethylene xide head rups. An alternate treatment 0 fr head rup interactins in such systems, cnsiderin the head rups t be plymerlike, has been develped by us, and it shwed mixed success in predictin the micellar prperties HEAD GROUP INTERACTIONS DIPOLAR If the surfactant has a zwitterinic head rup with a permanent diple mment, then diple diple interactins arise at the micelle surface. The diples are

15 Mlecular Thermdynamics f Giant Micelles 13 riented nrmal t the interface and stacked such that the ples f the diples are lcated n parallel surfaces. The interactin free enery is estimated by cnsiderin that the ples f the diples enerate an electrical capacitr and the distance between the planes f the capacitr is equal t the distance f chare separatin d (r the diple lenth) in the zwitterinic head rup. Cnsequently, ne ets fr spherical micelles 0 ( ) µ diple kt π e RS = εa kt δ d R + δ+ d S (1.0) where e is the electrnic chare, ε the dielectric cnstant f the slvent, R S the radius f the spherical cre, and δ the distance frm the cre surface t the place where the diple is lcated. This equatin is emplyed als fr lbular micelles and the endcaps f rdlike micelles. Fr the cylindrical part f the rdlike micelles, the capacitr mdel yields ( ) µ diple kt π e R d C = ln 1 + εakt δ R + δ C (1.1) where R C is the radius f the cylindrical cre f the micelle. The dielectric cnstant is taken t be that f pure water 44 and is calculated usin the expressin 0 ε = exp[ ( T 73)] (1.) HEAD GROUP INTERACTIONS IONIC If the surfactant has an aninic r catinic head rup, then inic interactins arise at the micellar surface. An apprximate analytical slutin t the Pissn ltzmann equatin derived by Evans and Ninham 45 fr spherical and cylindrical micelles is used in ur calculatins. ( ) µ inic kt ln S = + U S ( U 1) C + S ln 1 1 U κ S U = 1+ / 1 4 π e, S = εκ δ a k T (1.3) In Equatin 1.3, the area per mlecule a δ is evaluated at a distance δ frm the hydrphbic cre surface (see Table 1.1) where the center f the cunterin is

16 14 Giant Micelles: Prperties and Applicatins lcated. The first tw terms n the riht-hand side f Equatin 1.3 cnstitute the exact slutin t the Pissn ltzmann equatin fr a planar emetry and the last term prvides the curvature crrectin. The curvature-dependent factr C is iven by 0 1 C =,, R + δ R + δ R + δ S eq C (1.4) fr spheres/spherical endcaps f sphercylinders, lbular areates (with an equivalent radius R eq defined in Table 1.1) and cylindrical middle part f sphercylinders, respectively. κ is the reciprcal Debye lenth and is related t the inic strenth f the slutin via 8 π ne κ = ε kt 1/, ( C + C 1 a n = dd ) N Av 1000 (1.5) In equatin 1.5, n is the number f cunterins in slutin per cm 3, C 1 is the mlar cncentratin f the sinly dispersed surfactant mlecules, C add is the mlar cncentratin f any salt added t the surfactant slutin, and N Av is Avadr s number. 1.4 MODEL PREDICTIONS OF GIANT MICELLES COMPUTATIONAL APPROACH The equatin fr the size distributin f areates, in cnjunctin with the emetrical relatins in Table 1.1 and the expressins fr the varius cntributins t the free enery f micellizatin summarized in Sectin 1.3, allw us t make predictive calculatins f micellizatin behavir. The calculatins fr spherical and lbular micelles are carried ut based n the maximum-term methd 0,46 instead f cmputin the detailed distributin f sizes. This apprach is built n the recnitin that fr spherical r lbular micelles the size dispersin is usually narrw and n and w are practically the same. n can be taken as the value f fr which the number cncentratin X f the areates is a maximum, while w can be taken as the value f fr which the weiht cncentratin X f the areates is a maximum. Only the areatin number is the independent variable in the case f spherical r lbular micelles, and the maximizatin f either X r X can be carried ut with respect t. The cmc is estimated as the value f X 1 fr which the amunt f surfactant in the micellized frm is equal t that in the sinly dispersed frm. cyl In the case f rdlike micelles, by minimizin µ fr the cylindrical part, the equilibrium radius R C f the cylindrical part f the micelle is determined.

17 Mlecular Thermdynamics f Giant Micelles 15 Given the radius f the cylindrical part, the number f mlecules cap in the spherical endcaps is fund t be that value which minimizes µ. Given cap cap, µ cyl, and µ cap, the sphere-t-rd transitin parameter K is calculated frm Equatin 1.9, the averae areatin numbers at any ttal surfactant cncentratin frm Equatin 1.13, and the cmc frm Equatin The search fr the parameter values that maximize X r minimize µ cyl and µ cap was carried ut usin the IMSL (Internatinal Mathematical and Statistical Library) subrutine ZXMWD. This subrutine is desined t search fr the lbal extremum f a functin f many independent variables subject t any specified cnstraints n the variables. This subrutine has been used fr all the calculatins described in this chapter ESTIMATION OF MOLECULAR CONSTANTS Illustrative calculatins in this chapter have been perfrmed fr a few surfactant types. The mlecular cnstants assciated with the surfactant tail are the vlume v S and the extended lenth R S f the tail. Fr the head rup, ne needs the crsssectinal area a p fr all types f head rups, the distance δ frm the cre surface where the cunterin is lcated in the case f inic head rups, the diple lenth d, and the distance δ frm the cre surface at which the diple is lcated, in the case f a zwitterinic head rup. All these mlecular cnstants can be estimated frm the chemical structure f the surfactant mlecule Estimatin f Tail Vlume v S The mlecular vlume v S f the tail cntainin n C carbn atms is calculated frm the rup cntributins f (n C 1) methylene rups and the terminal methyl rup. The rup mlecular vlumes are estimated frm the density vs temperature data fr aliphatic hydrcarbns. 47 v = v + ( n 1) v S CH3 C CH v = ( T 98) nm CH3 3 (1.6) 5 v = ( T 98) nm 3 CH Estimatin f Extended Tail Lenth l S The extended lenth f the surfactant tail l S at 98 K is calculated usin a rup cntributin f nm fr the methylene rup and nm fr the methyl rup. 1,0 Given the small vlumetric expansin f the surfactant tail ver the rane f temperatures f interest, the extended tail lenth l S is cnsidered as temperature independent and the small vlumetric expansin is accunted fr by increases in the crss-sectinal area f the tail.

18 16 Giant Micelles: Prperties and Applicatins TALE 1. Mlecular Cnstants fr Surfactant Head Grups Surfactant Head Grup a p (nm ) a (nm ) δ (nm) d (nm) β-glucside Dimethylphsphenexide Sdium sulfate Trimethylammnium brmide n-alchl Estimatin f Head Grup Area a p The head rup area a p is calculated as the crss-sectinal area f the head rup near the hydrphbic cre water interface. The lucside head rup in β-lucsides has a cmpact rin structure 16,9 with an apprximate diameter f 0.7 nm, and hence, the effective crss-sectinal area f the plar head rup a p is estimated as 0.40 nm. Fr sdium alkyl sulfates, alkyl trimethyl ammnium brmides, alkyl dimethyl phsphene xides and n-alchls, the crss-sectinal area f the plar rup a p has been estimated takin the bnd lenths and bnd anles int accunt as equal t 0.17, 0.54, 0.48, and 0.08 nm, respectively (see Table 1.) Estimatin f δ and d Fr inic surfactants, the mlecular cnstant δ depends n the size f the inic head rup, the size f the hydrated cunterin, and als the prximity f the cunterin t the chare n the surfactant in. Visualizin that the sdium cunterin is placed n tp f the sulfate anin, we estimate δ = nm fr sdium alkylsulfates. Fr alkyltrimethyl ammnium brmide, the surfactant catin is nearer the hydrphbic cre, and we estimate δ = nm. All the mlecular cnstants characterizin the head rups f surfactants cnsidered in this chapter are listed in Table 1.. It is imprtant t nte that while the mlecular thery bviusly needs many mlecular cnstants, there are n free parameters invlved in this thery. N infrmatin frm experimentally determined prperties f surfactant slutins has been used t deduce any mlecular cnstant. Hence, the calculatins are cmpletely predictive in nature. Mst f the required mlecular cnstants such as dielectric cnstant, surface tensin, mlecular vlumes, and s frth, are already well knwn in the literature. Fr all practical purpses, we need t estimate nly ne mdel parameter fr nninic surfactants (a p ), tw fr inic surfactants (a p and δ) and three fr zwitterinic surfactants (a p, δ, and d). Clearly any perceptin that a mlecular theretical apprach requires many parameters is entirely incrrect.

19 Mlecular Thermdynamics f Giant Micelles Octyl lucside Micelle hydrdynamic radius (nm) a p = 0.39 nm a p = 0.40 nm Cncentratin f ctyl lucside (mm) FIGURE 1. Dependence f the micellar size (expressed as the hydrdynamic radius) n the cncentratin f the surfactant fr nninic ctyl lucside. th lines crrespnd t predicted values but fr marinally different values f the mlecular cnstant a p. Squares dente reprted experimental data. 48 Circles crrespnd t mdified experimental data if the reprted hydrdynamic radius had included ne layer f water. See text fr discussin ILLUSTRATIVE PREDICTED RESULTS In Fiure 1., the predicted hydrdynamic radius f nninic ctyl β-lucside micelle is pltted aainst the surfactant cncentratin. The hydrdynamic radius f the micelle is calculated frm the predicted weiht-averae areatin number usin the expressin R H v w S = 3 4 π 1/3 + l p (1.7) where l p is the lenth f the plar head rup which fr the rin structure f the β-lucside has been estimated 16,9 t be 0.7 nm. One may nte frm Equatin 1.9 that the sphere-t-rd transitin parameter K, which determines w (see Equatin 1.13), can be dramatically altered by small chanes in the free enery difference ( µ cap µ cyl ), since cap that appears in the definitin f K is much larer than unity. Fr example, assumin a typical value f 90 fr cap, a small chane f 0.05 k T in the free enery difference ( µ cap µ cyl ) will cause a chane in K f e 4.5 = 90, which in turn can chane the predicted value fr w by

20 18 Giant Micelles: Prperties and Applicatins a factr f abut 10. Therefre, the predicted averae areatin numbers are very sensitive even t small chanes in the free enery estimates when rdlike micelles frm. This is illustrated by the calculatins carried ut fr tw marinally different values f the parameter a p, which affects the manitude f the head rup steric interactin enery. The predictins are cmpared with the data prvided by dynamic liht scatterin measurements. 48 It is nt clear whether the experimental hydrdynamic radii crrespnd t dry r hydrated areates. Therefre, bth the reprted hydrdynamic radii and the radii btained by subtractin the diameter f a water mlecule are pltted in Fiure 1.. Given the sensitivity f K t the free enery estimates, the areement between the measured and predicted areate sizes is satisfactry. We nte that althuh rdlike micelles are frmed, they are still nt very lare fr ctyllucside. When the alkyl chain lenth increases, cap will increase and, cnsequently, K will increase sinificantly, leadin t the frmatin f iant rdlike micelles. The predicted values fr the sphere-t-rd transitin parameter K fr aninic sdium ddecyl sulfate as a functin f the added NaCl cncentratin are pltted in Fiure 1.3 and the predictins are cmpared aainst results frm liht scatterin measurements. 49 The larest deviatin between predicted and experimental value f K is less than a factr f 10 at the hihest inic strenth. As already nted, Sphere-t-rd transitin parameter K SDS, 5 C Experimental Predictins Cncentratin f added NaCl (M) FIGURE 1.3 Dependence f the sphere-t-rd transitin parameter K fr sdium ddecyl sulfate n the cncentratin f added electrlyte NaCl. Pints are frm liht scatterin measurements 49 at 5 C and predictins are shwn with the same symbls (in smaller size) and the cnnectin line.

21 Mlecular Thermdynamics f Giant Micelles 19 Weiht averae micelle areatin number w SDS, 0.8M NaCl 5 C Spheres Rds X FIGURE 1.4 Predicted weiht-averae areatin numbers f cexistin ppulatins f lbular micelles and cylindrical micelles as functins f the sinly dispersed SDS cncentratin. Calculatins are fr 0.8 M added NaCl and 5 C. Fr cncentratins near the cmc, the micelles are relatively small and lbular, while at hiher cncentratins, the rdlike micelles shw sinificant rwth in size. small variatins in the free enery estimates f the rder f 0.05 k T are adequate t cause a variatin in K by a factr f 100. Cnsiderin this, the deviatins between experimental data and predictins can be cnsidered satisfactry. The predicted radius f the cylindrical part f the areate is smaller than the fully extended lenth f the surfactant tail. It increases frm 1.45 nm t 1.49 nm as the electrlyte cncentratin is increased frm 0.45 M t 1.5 M at 5 C. Fr this rane f inic strenths, the radius f the endcap remains unaltered and is equal t the extended lenth f the surfactant tail. The predicted cmc, as defined by Equatin 1.14, decreases frm 0.38 mm t 0.04 mm ver this rane f added salt cncentratin. When rdlike micelles frm, smaller spherical/lbular micelles can als cexist. The relative cncentratin f these tw ppulatins f areates determines the averae slutin prperties f the surfactant system. Fiure 1.4 presents the weiht-averae areatin numbers crrespndin t bth smaller lbular micelles and larer cylindrical micelles frmed in slutins f sdium ddecyl sulfate at 0.8 M added NaCl. The micelle averae areatin numbers are pltted aainst the cncentratin f the sinly dispersed

22 0 Giant Micelles: Prperties and Applicatins SDS, 0.8 M NaCl, 5 C Spheres Rds ΣX X 1 FIGURE 1.5 Predicted amunt f surfactant incrprated int cexistin ppulatins f lbular micelles and cylindrical micelles as a functin f the sinly dispersed SDS cncentratin. Calculatins are fr 0.8 M added NaCl and 5 C. Fr cncentratins near the cmc, the lbular micelles dminate, while at hih cncentratins the rdlike micelles dminate. surfactant (which cntrls the chemical ptential f the surfactant in slutin). In Fiure 1.5, the amunt f surfactant incrprated int the lbular micelles and that in the rdlike micelles are pltted, als aainst the sinly dispersed surfactant cncentratin. Clearly, in the rein near the cmc and belw, the lbular micelles dminate. Just beynd the cmc, the lare rdlike micelles dminate. The temperature dependence f the sphere-t-rd transitin parameter K has been calculated fr sdium ddecyl sulfate fr tw cncentratins f added NaCl. The predicted values f K are pltted in Fiure 1.6 aln with the experimental values determined frm dynamic liht scatterin measurements. 49 The larest deviatin in K is abut a factr f 100, and the thery predicts a smewhat strner dependence f K n the temperature than that bserved experimentally. Fiure 1.7 presents the predicted values f K as a functin f added electrlyte cncentratin fr the hmlus family f sdium alkyl sulfates with 10 t 13 carbn atms in their hydrphbic tails. The fiure shws that the larest deviatin in K between the predicted and the experimental values is smaller than a factr f 5 fr C 11, C 1, and C 13 sdium sulfates, and is abut a factr f 100 fr the C 10 sdium sulfate.

23 Mlecular Thermdynamics f Giant Micelles 1 Sphere-t-rd transitin parameter K SDS 0.45 M NaCl, experiment 0.80 M NaCl, experiment 0.45M NaCl, predictins 0.80 M NaCl, predictins Temperature C FIGURE 1.6 Influence f temperature n the sphere-t-rd transitin parameter K f sdium ddecyl sulfate in slutins cntainin 0.45 M and 0.80 M added NaCl electrlyte. Pints are experimental data 49 and the predictins are shwn with the same symbls (in smaller size) and the cnnectin line Sdium Alkyl sulfates, 30 C Sphere-t-rd transitin parameter K C 13 C 1 C 11 C Cncentratin f added NaCl (M) FIGURE 1.7 Dependence f the sphere-t-rd transitin parameter K n the surfactant tail lenth and n the cncentratin f added electrlyte NaCl fr sdium alkyl sulfates. Pints dente the experimental data 49 at 30 C, and predictins are shwn with the same symbls as the crrespndin experimental pints (in smaller size) and the cnnectin line.

24 Giant Micelles: Prperties and Applicatins 1.5 TUNING STRUCTURAL TRANSITIONS IN GIANT MICELLES SURFACTANT MIXTURES Transfrmatin f ther areate structures t rdlike micelles and vice versa is pssible by the additin f a secnd surfactant. Such additin can cause all slutin prperties f the surfactant such as the cmc and micelle size, as well as the areate shapes t chane. We have frmulated a mlecular thery fr surfactant mixtures t predict the wide rane f variatins caused by different cmbinatins f binary surfactant mixtures. 18,1 The treatment is a direct extensin f the mlecular thery develped fr a sinle surfactant. Fr binary surfactant mixtures, the size distributin f areates has the frm µ µ µ µ A X = X X A 1A 1 exp A = X X exp 1A 1 1A 1 kt kt (1.8) The mle fractins f A and in the sinly dispersed surfactant mixture are dented as α 1A and α 1 while the mle fractins f A and in the mixed areates are dented by α A and α. Expressins fr the standard free enery chane n areatin are btained by a simple extensin f the equatins develped in Sectin 1.3 fr sinle cmpnent surfactant systems. Only the mdificatins necessary fr the treatment f surfactant mixtures are described belw. Mre details can be fund in ur earlier papers Free Enery f Frmatin f Mixed Micelles Transfer f the Surfactant Tail Fr a mixed micelle f cmpsitin (α A, α ), the transfer free enery per surfactant mlecule is the weihted averae f the cntributins frm the tw different tails. ( ) = ( ) ( ) tr, kt µ µ µ tr tr,a α + α A kt kt (1.9) This transfer free enery des nt include cntributins frm the mixin f A and tails inside the micellar cre, which are accunted fr separately Defrmatin f the Surfactant Tail If surfactants A and have different tail lenths, prtins f bth mlecules may nt be simultaneusly present everywhere in the micellar cre. Let us assume

25 Mlecular Thermdynamics f Giant Micelles 3 that surfactant A has a lner tail than surfactant, l SA > l S. If the micelle radius R is less than bth l SA and l S, then even the shrter tail can reach everywhere within the cre f the micelle. If l SA > R S > l S, then the inner rein f the micellar cre, f dimensin (R S l S ), can be reached nly by the A tails. Takin int accunt the different extent t which the A and the tails are stretched fr the tw situatins described abve, we have btained the expressin ( µ ) R def S αa kt = N L + A α Q = N L 9 pπ, 80 Q = R if R l, l Q = l if l R l s A S S S S SA S S (1.30) N A and N stand fr the number f sements in the tails f surfactants A and, respectively, and p is the packin factr defined in Table 1.1. This equatin is used fr spherical and lbular micelles and fr the spherical endcaps f rdlike micelles. Fr the cylindrical part f the rdlike micelles, the cefficient 9 in is replaced by 10, the radius R S is replaced by the radius R C f the cylindrical cre, and the packin parameter p = 1/ Frmatin f Areate Cre Water Interface The free enery assciated with the frmatin f an interface is iven by the expressin ( µ ) int kt = σa ( kt a α a α a ) A A (1.31) Here, a A and a are the areas per mlecule f the cre surface shielded frm cntact with water by the head rups f surfactants A and. Since the interfacial tensin aainst water f varius hydrcarbn tails f surfactants are clse t ne anther, σ a is apprximated by the micelle cmpsitin averaed value Head Grup Steric Interactins Extendin the expressin used fr sinle surfactant systems t binary surfactant mixtures, ne can write ( ) ln 1 kt = µ α a + α a steric A pa p a (1.3)

26 4 Giant Micelles: Prperties and Applicatins Head Grup Diple Interactins The diple diple interactins are cmputed fr spherical and lbular micelles and the spherical endcaps f rdlike micelles frm ( ) µ diple kt π e RS = ε a k T diple d α R + δ + d S,diple (1.33) Fr the cylindrical part f the rdlike micelles, ( ) µ diple kt π e RC = εa k T diple ln d α 1 +,diple R C + δ (1.34) In the precedin relatins, α,diple is the fractin f surfactant mlecules in the areate havin a diplar (zwitterinic) head rup. If bth A and are zwitterinic, then α = α + α = 1, a = a,diple A diple δ (1.35) If surfactant A is zwitterinic and surfactant is nninic r inic, then α a = α, a = α,diple A diple δ,diple (1.36) The diple diple interactins may arise even if neither f the surfactants is zwitterinic. Such a situatin ccurs when the surfactant mixture cnsists f an aninic and a catinic surfactant. The tw ppsitely chared surfactants may be visualized as frmin in pairs. Dependin upn the lcatin f the chares n the tw surfactant head rups, these in pairs may be assined a diple mment. The distance f chare separatin d in the zwitterinic head rup, nw refers t the distance between the lcatins f the aninic and the catinic chares, measured nrmal t the micelle cre surface. Thus, fr such systems α α α A the smaller f (, ), =,diple diple a a δ =, d = δ δ A α,diple (1.37) The factr in the expressin fr α, diple accunts fr the fact that a diple is assciated with tw surfactant mlecules, treated as a pair. δ A and δ represent the distance nrmal t the hydrphbic cre surface at which the chares are lcated n the A and surfactants.

27 Mlecular Thermdynamics f Giant Micelles Head Grup Inic Interactins The inic interactins at the areate surface are calculated usin Equatin 1.3, in cnjunctin with the curvature factr C iven by Equatin 1.4, with the mdificatin that S is nw iven by 4π e aδ S =, a =, δ= α δ δin εκa α +α δ A A kt δ in in (1.38) α in = 1, if A and are inic f the same kind α = α, δ = 0, if nly A is inic and is nninic r zwitterinic in A α = α α, if A and are inic and ppsitely chared in A Free Enery f Mixin f Surfactant Tails This is the nly cntributin that is nt present in the free enery mdel fr sinle cmpnent surfactant slutins. This cntributin accunts fr the entrpy and the enthalpy f mixin f the surfactant tails f A and in the hydrphbic cre f the micelle, with respect t the reference states f pure A r pure micelle cres. This cntributin is calculated usin the Flry Huins expressin. 50 ( µ ) mix = α η + α η A A kt [ ln ln ] + α A v ( ) + ( ) δ δ α v δ δ H mix H H H SA A mix S kt (1.39) H H δ = η δ + η δ mix A A H H H where η A and η are the vlume fractins f A and in micelle cre, δa and δ are the Hildebrand slubility parameters 51 fr the tails f surfactants A and, H and δmix is the vlume fractin averaed slubility parameter f the tw cmpnents within the micelle cre Estimatin f Mlecular Cnstants and Cmputatinal Apprach The nly mlecular cnstant that is new fr calculatins invlvin surfactant mixtures is the Hildebrand slubility parameter fr the hydrcarbn tails f the surfactants. This can be estimated usin a rup cntributin apprach based n the prperties f pure cmpnents. 51,5 Fr aliphatic hydrcarbn tails, the slubility parameters can be estimated frm ( n 1) δ = v H C 1/ S MPa, 3 J v S in nm, 1MPa = 1 cm 3 (1.40)

28 6 Giant Micelles: Prperties and Applicatins The cmputatinal apprach fllws the descriptin in Sectin Fr the binary mixtures, the cncentratin f the sinly dispersed surfactant mixture X 1 (= X 1A + X 1 ) and the cmpsitin f this mixture α 1A are used as inputs Predicted Results Calculatins have been carried ut fr binary mixtures f aninic sdium ddecylsulfate (SDS) and nninic decyldimethylphsphenexide (C 10 PO) at 5 C with 0.8 M added NaCl. Results fr pure SDS presented in Sectin shw that SDS frms lare rdlike micelles with K ~ at this cnditin. In cntrast, pure C 10 PO at this cnditin is predicted 1 t frm small, lbular micelles with areatin number f abut 50. The tw surfactants have slihtly different lenths f hydrphbic tails, the sizes f their head rups are different, and ne head rup is inic. Hence the prperties f mixed areates will depend strnly n the cmpsitin. The predicted cmc f the mixture is pltted in Fiure 1.8 as a functin f bth the micelle cmpsitin and the cmpsitin f the sinly dispersed surfactant. Frm this fiure we nte that the fractin f SDS in micelles is always larer than the fractin f SDS in the sinly dispersed surfactant. The predicted sphere-t-rd transitin cnstant K is pltted aainst the cmpsitin f the micelle and the cmpsitin f the sinly dispersed surfactant in Fiure 1.9. We find that n the additin f the nninic C 10 PO t the aninic SDS rdlike micelles, the rds rw (K increases) sinificantly 10 SDS + C 10 PO, 5 C α micelle α mnmer cmc (M) Fractin f SDS in micelles r mnmers FIGURE 1.8 cmc f SDS + C 10 PO binary mixture as a functin f the cmpsitin f micelles (circles) and that f sinly dispersed surfactants (trianle) at 5 C. Fractins refer t mle fractin in the binary surfactant mixture. th pints and lines are predicted values.

29 Mlecular Thermdynamics f Giant Micelles 7 Sphere-t-rd transitin parameter K SDS + C 10 PO, 5 C α micelle α mnmer Fractin f SDS in micelles r mnmers FIGURE 1.9 Sphere-t-rd transitin parameter K fr SDS + C 10 PO binary mixture as a functin f the cmpsitin f micelles (circles) and that f sinly dispersed surfactants (trianle) at 5 C. th pints and lines are predicted values. initially, reach a maximum in size, and then decrease in size, eventually transfrmin int small lbular micelles. This behavir can be interpreted in terms f the varius interactins. The head rup f SDS (0.17 nm ) is smaller in size cmpared t the head rup f C 10 PO (0.48 nm ). When small amunts f nninic C 10 PO are incrprated in the SDS micelles, the inic head rup repulsins at the micelle surface decrease withut at the same time increasin the steric repulsins sinificantly. In the free enery mdel, this implies the lwerin f bth µ cyl and µ cap, but the lwerin f µ cyl is larer in manitude cmpared t the lwerin f µ cap, cntributin t an increase in K. When larer amunts f C 10 PO are incrprated, the steric interactins becme mre sinificant. Under these cnditins, the lwerin f µ cyl is smaller in manitude cmpared t the lwerin f µ cap, cntributin t a decrease in K. When the fractin f SDS in the micelle becmes smaller than abut 0.50, the rds cease t exist and micelles becme lbular. Thus we see bth a rwth f rds and a rd-t-sphere transitin n the additin f a nninic surfactant. Fiure 1.10 shws the predicted areatin numbers f SDS-C 10 PO micelles as a functin f micelle cmpsitin when the ttal surfactant cncentratin is in the prximity f the cmc and when the micelles are small and lbular. The dependence f the size f these small micelles n micelle cmpsitin shws a maximum similar t the dependence f K n micelle cmpsitin. When the nninic mlecules are added t the inic micelle, the resultin reductin in inic