Micelle formation of sodium cholate and solubilization into the micelle

Hiromi Sugioka, Yoshikiyo Moroi *Department of Chemistry, Faculty of Science, Kyushu University, Higashi-ku, Fukuoka 812-8581, Japan

Received 1 May 1998; accepted 13 May 1998

Abstract

The micellization of sodium cholate (NaC) was studied at 298.2 K by aqueous solubility at different pH values. Using astepwise association model of cholate anions without the sodium counterion, the aggregation number (n) of the cholatemicelle was evaluated and found to increase with the total concentration, indicating that the mass action model worked quitewell. The n value at 60 mM was found equal to 16. The membrane potential measurement of sodium ion with a cationexchange membrane was made in order to confirm the low counterion binding to micelle. The solubilization of alkylbenzenes(benzene, toluene, ethylbenzene, n-propylbenzene, n-butylbenzene, n-pentylbenzene, n-hexylbenzene) and polycyclic aromaticcompounds (naphthalene, anthracene, pyrene) into the aqueous micellar solution of sodium cholate was carried out.Solubilizate concentrations at equilibrium were determined spectrophotometrically at 298.2 K. The first stepwise associationconstants (K1) between solubilizate monomer and vacant micelle were evaluated from the equilibrium concentrations andfound to increase with increasing hydrophobicity of the solubilizate molecules. From the Gibbs energy change forsolubilization at the different mean aggregation numbers and from molecular structure of the solubilizates, the function ofsodium cholate micelle for solubilization was discussed and was compared with data from conventional aliphaticmicelles. ß 1998 Elsevier Science B.V. All rights reserved.

Keywords: Sodium cholate; Micellization; Aqueous solubility; Aggregation number; Alkylbenzene; Polycyclic aromatic compound;Solubilization

1. Introduction

Bile salts, having a large, rigid, and planar hydro-phobic moiety of a steroid nucleus with two or threehydroxyl groups, are a special group of biosurfac-tants, whose properties di¡er considerably from or-dinary aliphatic surfactant molecules [1]. They act assolubilizer and emulsi¢er for cholesterol and lipids inthe intestines. Because of the interesting physiologi-cal functions, micellar properties of bile salts have

been extensively studied by various methods [2^16],and many review articles on physicochemical proper-ties of bile have appeared [17].

Bile salts have lower aggregation number com-pared with conventional aliphatic surfactants. Forthe sodium cholate micelle, lower values of 4 [2,3],and higher values of 16 [4,5], have been reportedfrom di¡erent measurements. The aggregation modelsuggested by Small was the primary^secondary mi-celle model. In this model, the primary micelles areformed in such a way that the hydrocarbon backs ofthe steroid nucleus associate. The secondary micellesare then formed by the aggregation of these primarymicelles [1]. This model invokes a stepwise aggrega-

0005-2760 / 98 / $ ^ see front matter ß 1998 Elsevier Science B.V. All rights reserved.PII: S 0 0 0 5 - 2 7 6 0 ( 9 8 ) 0 0 0 9 0 - 3

* Corresponding author. Fax: +81 (92) 642-2607;E-mail : moroiscc@mbox.nc.kyushu-u.ac.jp

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tion mechanism, i.e. polydispersity in the aggregates,in which the critical micelle concentration (CMC)appears not as a point but over a certain range.This fact was supported by other experimental dataderived from surface tension [6] and £uorescence [7]measurements. However, micelle formation takesplace in the bulk. In this sense, surface tension, bywhich the state of surfactant molecules on the sur-face is observed, is essentially an unsuitable methodfor investigating physicochemical properties of mi-celles in the bulk. Although it can be used, just forCMC determination. The £uorescence probe methodis convenient to study molecular aggregation in abulk, but the method is not totally recommendedin the sense that some other chemicals need to beadded in the system. On the other hand, the presentsolubility method depends just on thermodynamicswithout any other chemicals. In the stepwise aggre-gation model based on thermodynamics, the aggre-gation number should increase with amphiphile con-centration. However, very few publications giveinformation on this very important point.

The enhanced aqueous solubility of otherwiseslightly soluble organic substances brought aboutby the presence of surfactant micelles is well knownas solubilization [18,19]. This process, caused by in-corporation of hydrophobic organic substances intomicelles, plays a very important role not only in in-dustrial processes, but also in biological processes[20], such as adsorption and transfer of materials inliving tissues.

A function of bile salts is to solubilize sparinglysoluble substances such as cholesterol and bilirubininto bile. The solubilization of bile salts has beenextensively studied; for example, the solubilizationof phospholipid vesicles [21], and of aromatics[8,22^26] and the solubilization into mixed micelles[27]. Most of the studies were carried out under var-ious conditions in the presence of added salts in buf-fer solution. However, few basic and systematic stud-ies have been carried out. Mukerjee and Cardinalhave reported on the solubilization of naphthaleneinto sodium cholate micelles [22]. According tothem, if the CMC is estimated by the usual proce-dure of ¢nding the intercept from linear extrapola-tion of the data below and above the CMC, very

di¡erent CMC values are obtained. The CMC de-pends upon which sections of the data are used toderive the interpolated trace. Therefore, it is quitedi¤cult to precisely determine the CMC value ofbile salts. Hydrophobic solutes with a rigid structure,such as bile salts are expected to have rather an ex-tended concentration range over which the molecularaggregation number increases from low to high val-ues. The physicochemical studies of bile salts, on theone hand, and their physiological function as solubi-lizer on the other hand, make it clear that their be-havior in vitro and their functions in vivo are closelyrelated. In this sense, thermodynamic research on thesolubilization by bile salts is quite important.

Unfortunately, solubilization has been treated inmost cases as a partitioning of solubilizate moleculesbetween a micellar phase and an intermicellar bulkphase [28^31], which is inconsistent with the phaserule as pointed out by one of the authors [32,33]. Ifthe micelles are regarded as a phase, the number ofdegrees of freedom in the presence of a coexistingsolubilizate phase is two. This would mean that thesurfactant concentration could not be changed for agiven temperature and pressure. However, the max-imum additive concentration changes with total sur-factant concentration. So, it is preferable to analyzesolubilization data in terms of the thermodynamicparameters of solubilization, which regards a micelleas a chemical species [34,35].

In this paper, a proven method based on aqueoussolubility was applied for investigating the micelliza-tion of sodium cholate in water at 298.2 K. Thisthermodynamic method is very simple and usefulfor aggregates whose aggregation number is rela-tively small and increases with concentration. Theaim of the present study is to clarify the dependenceof aggregation number of cholate micelle on the con-centration using the method mentioned above. Thesecond objective of this study is to investigate thesolubilization of alkylbenzenes and polycyclic aro-matic compounds into sodium cholate micelles at298.2 K by the mass action model. This work servesto clarify the e¡ect of molecular structure of solubi-lizate and surfactant on the thermodynamic param-eters of solubilization and the characteristics of sol-ubilization by sodium cholate.

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2. Materials and methods

2.1. Materials

Cholic acid (CA) was of guaranteed reagent grade(Nacalai Tesque). The cholic acid was puri¢ed byfoam fractionation [36,37] of the sodium salt solu-tion around the critical micellar concentration,where more than 15% of the original solution wasremoved with foam. After foam fractionation, theresidual solution was acidi¢ed with dilute HCl sol-ution to precipitate the cholic acid from the solu-tion. The precipitate was washed with largeamounts of doubly distilled water and dried overP2O5 in a desiccator under reduced pressure. Thepurity was checked by elemental analysis, and theobserved and calculated values were in satisfactoryagreement CAW1/2H2O: C 68.83 (69.02), H 9.91(9.90)%, where the values in parentheses are thecalculated ones. The stock solution of NaC wasprepared by stoichiometric neutralization of cholicacid by sodium hydroxide solution and was dilutedto a ¢nal volume with doubly distilled water. Thesolution was ¢ltered through a 0.2 Wm membrane¢lter (Millipore FGLP01300) to remove dust fromthe solution. Cholate solutions for analytical usewere prepared by volumetric dilution of this stocksolution.

Benzene, of guaranteed reagent grade (NacalaiTesque), was used as received. Toluene and ethylben-zene of guaranteed reagent grade (Nacalai Tesque)were washed four times with concentrated sulfuricacid, four times with diluted sodium hydroxide sol-ution alternately, and ¢nally three times with water.n-Propylbenzene, n-butylbenzene, n-pentylbenzene,and n-hexylbenzene (Tokyo Kasei Kogyo) were dis-tilled once and rinsed with a large amount of water10 times [38].

Naphthalene, anthracene of guaranteed reagentgrade (Nacalai Tesque), and pyrene of similar grade(Tokyo Kasei Kogyo) were puri¢ed by repeated re-crystallizations from the ethanol solution [39]. Thepurity of these compounds was checked by elementalanalysis. Ethanol of 99.5 vol% (Nacalai Tesque) wasused without further puri¢cation.

Water for use was distilled twice from alkalinepermanganate solution.

2.2. Relationship between concentration and pH

The cholic acid was dispersed in 15 cm3 of doublydistilled water in a glass vessel. The suspension ofsolid cholic acid was agitated with a rotor while ther-mostatically controlled at 298.2 K. The pH of thesuspension was monitored with a pH electrode (Ho-riba), during the stepwise introduction of aliquots ofa dilute solution of sodium hydroxide. The total acidconcentration (Ct) was evaluated from the titrationwith an added amount of sodium hydroxide [40].The neutralization reaction of the acid with sodiumhydroxide is very rapid, and therefore, the concentra-tion of cholate anion should be equal to Na� concen-tration, even when aqueous solubility of undissoci-ated cholic acid would be below the saturation limit.

2.3. Membrane potential measurement

The measurements were made for sodium ion witha cation-exchange membrane (Tokuyama, Neocepta-CM1) that had been dipped in 1 mM NaC solutionfor a few days in order to replace all the exchange-able ions with sodium ion. The membrane potentialat 298.2 K was measured during the stepwise intro-duction of aliquots of a NaC solution into the sam-ple solution, using the following cell :

reference electrode (calomel) M agar bridge Mreference solution (C0 = 5 mM NaC) M cation-ex-change membrane M sample solution (Ct mMNaC) M agar bridge Mreference electrode (calomel)

where Ct denotes the varying NaC concentration.The electromotive force (vE) of the cell was meas-ured with a precision of þ 0.5 mV by using Advant-est digital multimeter (model R6441B).

2.4. Solubilization

2.4.1. Alkylbenzenes [38]A simple glass vessel having eight tubes (Fig. 1)

was used as an apparatus for the solubilization ofvolatile or gaseous substances. Eight surfactant sol-utions of di¡erent concentrations were poured sepa-rately into eight tubes of the glass vessel; the cholate

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concentration in the ¢rst four tubes was below theCMC and the last four were above the CMC ofsodium cholate. A minute amount of solubilizatewas placed on the hollow in the middle of the eight¢ngers. The glass vessel was set into a thermostatcontrolled at 298.2 K, while each of the surfactantsolutions was stirred for 24 h. After equilibration,each surfactant solution was separately drawn intoeach injection tube through an injection needle, andthe tubes were immediately capped with a siliconerubber. The optical absorbance of each solutionwas measured with an ultraviolet spectrophotometer(Hitachi model 100-50) in order to determine theequilibrium solubilizate concentrations from the mo-lar extinction coe¤cient. For n-pentylbenzene and n-hexylbenzene, approximately 1 cm3 of the liquid wasplaced on the hollow, because the aqueous solubilityis very small and their absorbance below the CMCcannot be observed unless their concentrations reachthe maximum additive concentration. However, forn-hexylbenzene, the aqueous solubility was below thedetection limit of spectrophotometer. The opticaldensity of n-hexylbenzene in pure water was calcu-lated to be ca. 0.001 using the molar extinction co-e¤cient of 150 mol31 dm3 cm31 and the aqueous

solubility of 5.98U1036 mol dm33. However, thelower detection limit of the spectrophotometer is ac-curate to 0.005. Therefore, we have used the extrapo-lated solubility value from the linear relationship be-tween the logarithm of aqueous solubility and thecarbon number of alkylchain of the solubilizates.

2.4.2. Polycyclic aromatic compounds [39,41]Four cm3 of sodium cholate solution and a small

amount of powdered solubilizate su¤cient to pro-duce a saturated solution were put together into a10 cm3 injector tube (Fig. 2). The solutions werestirred for about 24 h until equilibrium was reachedat 298.2 K, where the temperature was controlled towithin þ 0.005 K. Filtration of the excess solid wasperformed through a membrane ¢lter of 0.2 Wmpore-size by applying pressure upon the injector.The ¢ltrate was diluted by adding further surfactantsolution so that the absorbance of the diluted solu-tion becomes less than 0.7. The absorbance of thesample was measured spectrophotometrically, andthe maximum additive concentration (MAC) was de-termined using the respective molar extinction coef-¢cient given in Table 1.

3. Results

3.1. Micellization

Fig. 3 shows the relationship between the totalacid concentration of cholic acid, Ct, and pH. Ct isproportional to 1=aH� below the CMC of NaC [36]

Fig. 2. Solubilization apparatus for solid solubilizates.

Fig. 1. Solubilization apparatus for volatile solubilizates. Eightsurfactant solutions of di¡erent concentrations are poured sepa-rately into eight tubes. Inside the apparatus, the volatile solubi-lizates easily evaporate, and the chemical potential of the gas-eous solubilizate molecules becomes constant throughout all thephases.

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and departs from a straight line when molecular ag-gregation starts [40].

Fig. 4 displays the relationship between vE andsodium cholate concentration. The vE values de-crease linearly over the whole concentration range5^60 mM of NaC. All experimental points fall on astraight line over the whole concentration range

above and below the CMC. This strongly indicatesthat Na� concentration increases linearly with so-dium cholate concentration.

3.2. Solubilization

The solubilization data for n-butylbenzene and an-

Fig. 5. Change in concentration of n-butylbenzene with sodiumcholate concentration at 298.2 K.

Fig. 4. Variation of membrane potential change (vE) with so-dium cholate concentration at 298.2 K. A NaC solution of5 mM below the CMC is used as reference, and NaC concen-tration is measured over the range 5^60 mM. The CMC givenin the ¢gure is 12 mM from reference [37].

Fig. 3. Relationship between total cholate concentration andaqueous solution pH at 298.2 K. Monomer concentration at40 mM is determined by the way ACBCC. Point A, Ct =40 mM; point B, the value on the straight line at the same ab-scissa value of point A; point C, the ordinate value [S3] of point B.

Table 1Molar extinction coe¤cients (O) of solubilizates

Solubilizate O/mol31 dm3 cm31 Wavelength/nm

Benzene 1.25U102 253Toluene 2.17U102 260Ethylbenzene 1.83U102 260n-Propylbenzene 1.87U102 260n-Butylbenzene 1.79U102 260n-Pentylbenzene 1.64U102 260n-Hexylbenzene 1.50U102 260Naphthalene 4.92U103 276

(5.24U103)a (274)Anthracene 1.64U105 254

(1.49U105)a (250)Pyrene 4.02U104 273

(4.63U104)a (271)aThe values in parentheses are those in pure water and are usedfor determination of solubilizate concentration in NaC solutionbelow the CMC.

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thracene appear to be best represented by a smoothcurve, as Figs. 5 and 6 show respectively. Solubilizateconcentration remains almost constant for n-alkyl-benzenes and naphthalene below the CMC, whileincreasing slowly for anthracene and pyrene. Theaqueous solubility constants for the aromatics usedfor the thermodynamic analysis are from our pre-vious data [39,41].

4. Discussion

4.1. Micellization and micelle characterization

4.1.1. TheoryThe stepwise association of surfactant molecules

(S) for micellization, neglecting the counterion bind-ing to the micelle, can be expressed as

2S�b2

S2

3S�b3

S3

TTTTTT

nS�bn

Sn �1�

[32,42], where Ln is the equilibrium constant of mi-

celle formation expressed by concentration. Thecounterion binding to the micelle is assumed negli-gible, which is con¢rmed by the membrane potentialmeasurement in this work. This simpli¢cation of thepresent analysis seems reasonable. Thus, the totalanalytical concentration or the total equivalent con-centration of the molecules (Ct) and the total molarconcentration or the total concentration of osmoti-cally active particles (i.e. monomers+micelles) (Cm)become,

Ct �Xn

i�1

i�Si� �Xn

i�1

ibi�S�i �2�

Cm �Xn

i�1

�Si� �Xn

i�1

bi�S�i �3�

where [S] is the monomer concentration. Therefore,the following relation holds:

dCm=d�S� �Xn

i�1

ibi�S�i31 �Xn

i�1

�ibi�S�i=�S�� � Ct=�S�

�4�and then

Cm �Z �S�

0�Ct=�S��d�S� �5�

n � �Ct3�S��=�Cm3�S�� �6�where n is the mean micellar aggregation number.

Fig. 7. Change in monomer concentration of sodium cholateplotted against the total concentration at 298.2 K.

Fig. 6. Change in concentration of anthracene with sodiumcholate concentration at 298.2 K.

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Eq. 5 means that the total molar concentration canbe evaluated from the relationship between the totalanalytical concentration (Ct) and the monomericconcentration ([S]).

4.1.2. Determination of mean aggregation numberWhen an excess solid phase of a monobasic acid

(HS) coexists with an aqueous phase, the total solu-bility or the total concentration (Ct) is expressed as[43]

Ct � �HS� � �S3� �7�

� �HS� � ��HS�Ka=cS3�=aH� ; �8�

where the acidity constant (Ka) is given by

Ka � �S3�cS3aH�=�HS� �9�and the activity of the undissociated species (HS) isassumed equal to its concentration because it is ex-tremely low. From Eq. 8, Ct of cholic acid is propor-tional to 1=aH� below the CMC of NaC. AlthoughCt departs from a straight line above the CMC (Fig.3), the monomer concentration ([S3]) can be esti-mated from the solution pH by the linear extrapola-tion of the values below the CMC. This is only pos-sible as long as the acid phase coexists in the system[36]. For example, if Ct = 40 mM, construct the bro-ken line from point A for Ct = 40 mM on the curveto point B on the extrapolated line derived from theexperimental concentrations below the CMC. Theordinate value (point C) of point B gives the sumof undissociated cholic acid concentration ([HS])and monomer concentration ([S3]). The [HS] valueis so low, 0.122 mM at 298.2 K [36], that it can benegligible for the Ct values. Therefore, the ordinatecan be regarded as indicating the [S3] value. Thiswas con¢rmed in a similar previous study [40].

In this study, however, undissociated cholic acidmay disturb the micellization of cholate anions. For-tunately, the ratio of undissociated to dissociatedacid is less than 0.01 at the CMC, as estimatedfrom the aqueous solubility (0.122 mM) and theCMC value (12^15 mM). Such a small amount ofundissociated species was shown to have little in£u-ence on the CMC value in our previous study [40].Sodium cholate micelles solubilize undissociatedcholic acid in the form of acid^salt mixed micelles.To examine the extent of the solubilization of undis-

sociated cholic acid by the cholate micelle, the fol-lowing gravimetric analysis was performed. Centri-fuge tubes containing both NaOH solution of60 mM and excess coexisting solid of CA were placedinto a thermostat at 298.2 K for 24 h with continu-ous stirring. After centrifuging the suspension, thesupernatant was further ¢ltered with 0.2 Wm mem-brane ¢lter (Millipore FGLP01300) in order to re-move further small solid particles of CA completely.A ¢nite volume of the ¢ltrate was transferred to aweighing bottle, and was dried over P2O5 in a desic-cator under reduced pressure. The weight of the res-idue corresponding to the sum of dissolved NaC,undissociated CA, and CA solubilized into NaC mi-celles was measured. The solubilized CA was deter-mined by subtracting the dissolved NaC and solubil-ity of CA from the measured weight. The ¢nal resultis that the excess (solubilized) CA is 1.7% by weightof the calculated amount. This value means that themean number of CA per micelle is only 0.27 at aconcentration of 60 mM. Therefore, in this system,the small amount of solubilized undissociated acid isnot expected to change the intrinsic micellar proper-ties at all, as determined from this experimental factand other experimental results from mixed micelliza-tions [32,40]. The solubility below the CMC agreedwith the calculated value within the experimental tol-erances of a weighing balance.

Fig. 8. Plots of Ct/[S3] against [S3]. These plots are made inorder to determine the molar concentration Cm. The ordinaterepresents the integrand of Eq. 5.

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The monomer concentration change with totalacid concentration is shown in Fig. 7. Monomer con-centrations increase quite slowly with Ct above theCMC. Plots of Ct/[S3] vs. [S3] for the numericaldetermination of the Cm values by Eq. 5 are givenin Fig. 8. The molar concentration Cm can be calcu-lated by the integration of Ct/[S3] with respect to[S3], the calculation was made graphically from theevaluation of the area below the curve in Fig. 8.Thus, the change in integration limits of Ct/[S3]with respect to [S3] enables us to determine the nvalue at arbitrary concentrations of Ct from Eq. 6.It is obvious that the n value below the CMC is one,as the above analysis suggests, whereas the value ofCt/[S3] starts to deviate upward from unity abovethe CMC.

Thus, this thermodynamic method enables us todetermine the aggregation number change with con-centration by only pH measurement. This method isvery suitable for aggregates like bile salts whose ag-gregation number is relatively small and increaseswith concentration.

4.1.3. Dependence of the mean aggregation number onconcentration

In Fig. 9 the n values are shown plotted against thetotal concentration. The n value at 60 mM is ca. 16.This value agrees with those by Lindman et al. [3]and Fontell [2]. On the other hand, several authorsreported lower n values, 3^4 [4,5], but the depend-ence of aggregation number on the concentration hasnot been investigated in their studies. Anyhow, so-dium cholate micelles have smaller aggregation num-ber compared with conventional aliphatic surfactantmicelles. The n value increases with the total surfac-tant concentration, as the curved solid line in Fig. 9shows. This fact strongly indicates that the mass ac-tion model in the theoretical section, which regardsmicelles as a chemical species, is applicable. Lindmanet al. estimated the aggregation number from theself-di¡usion coe¤cient of solubilized decanol andgave information on the change of aggregation num-ber with concentration [3]. Their results agree wellwith ours. The n values in their work are little largerthan those found in this work, which seems to be dueto solubilized decanol. On the other hand, accordingto Coello et al. [13], the dependence of the aggrega-tion number on concentration is not observed over

the concentration range 0.0460 (about 45 mM) to0.205 (about 203 mM) mol kg31 .

Several investigators suggest that bile salts havethree limiting association concentrations [8,22]: limit1, 13^15 mM; limit 2, 45^50 mM; and limit 3, 90^110 mM. This is consistent with our data for sodiumcholate. According to the primary^secondary micellemodel proposed by Small [1], four molecules of bilesalts are regarded as the basic unit of aggregation.Judging from the present results, the aggregationnumber of sodium cholate seems to increase step-wise: 8, 16, and so forth.

At any rate, the proposed solubility method en-ables us to reasonably study micellization of systemswhose aggregation number increases with total con-centration without any assumption or any additive.This claim cannot be made from surface tension and£uorescence methods.

4.1.4. Degree of counterion binding to micelleIn the section of micellization theory, the counter-

ion is assumed not to take part in micellization. Toexamine whether this assumption is correct or not,the membrane potential measurement for sodium ionwith cation-exchange membrane was performed at298.2 K. Ideally, the relation between membrane po-tential vE and activity of sodium ion aNa� obeys theNernst's equation,

Fig. 9. Dependence of the mean aggregation number on the to-tal cholate concentration at 298.2 K. Mean aggregation numberwas evaluated from Eq. 6, using Ct, Cm, and [S3] values. Plotsare ¢tted to a smooth curve.

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vE � 3�RT=F�ln�aNa�=a0� � 359:2 log aNa� � const:;

�10�

where F is the Faraday's constant and a0 is the ac-tivity of reference solution. In this study, vE valuesare plotted against the logarithm of the sodium chol-ate concentration.

If counterion association to the micelle takes placeas to Na� ions in the bulk above the CMC, thestraight line must deviate upwards from the CMC.However, several other papers have reported thatthis is not the case for the sodium cholate, where alow degree of counterion binding is observed[2,13,15]. This fact implies that the counterion Na�

is only slightly bound to the micelle, which indicatesthat counterion binding to the micelle can be ne-glected. This fact was also substantiated by a linearrelationship between speci¢c conductivity and theconcentration of sodium cholate in solution. In addi-tion, if the total concentration exceeds 60 mM in Fig.3, the solid line starts to deviate towards the ordi-nate. This is an indication for the onset of counterionbinding [40]. In this region, the above theory cannotbe applied and a modi¢ed theory including the coun-terion binding should be used. In this paper, how-ever, only a lower concentration range was analyzedin order to avoid additional complexity for the mo-ment.

4.2. Solubilization of organic compounds

4.2.1. TheoryThe micellar aggregation number n is assumed to

be constant, i.e. micelles are monodisperse. This is toavoid the mathematical di¤culties arising from treat-ing them as polydisperse. Even in the case of thepolydispersity, the present discussion remains essen-tially valid [34].

Micelle formation equilibrium between surfactantmonomer (S) and micelles (M), whose mean aggre-gation number is n, is expressed as

nS�Kn

M �11�

where Kn is the equilibrium constant of micelle for-mation (Kn and M correspond respectively with Ln

and Sn in the theory section of Section 4.1.1).When the equivalent concentration of solubilizate

is less than a few times the micelle concentration,incorporation of the solubilizates into micelles is as-sumed not to change the intrinsic properties of theparent micelles. The stepwise association equilibriumbetween micelles and solubilizates (R) is

M�R�K1

MR1

MR1 �R�K2

MR2

TTTTTTTTTTTTTTT

MRm31 �R�Km

MRm �12�

where MRi is the micelle associated with i solubili-zate molecules, Ki is the stepwise association con-stant between MRi31 and a solubilizate monomer,and m is the maximum number of solubilizate mol-ecules per micelle. When an excess solid phase coex-ists with a surfactant solution phase, the maximumadditive concentration (MAC) is ¢xed by specifyingthe total surfactant concentration for a given con-stant temperature and pressure. This arises from adegree of freedom of three as de¢ned by the Gibbs'phase rule for this system. In the case for the alkyl-benzenes, the chemical potential can be kept constantthroughout all the phases, and then the K1 value canbe obtained in the same way as that for solid solu-bilizates.

The assumption described above can lead to thefollowing equations:

Kj � K1=j �13�

P�i� � Ri exp�3R�=i! �14�where P(i) is the probability that a micelle is associ-ated with i solubilizate molecules, R is the averagenumber of solubilizate molecules per micelle. Eq. 14is the Poisson distribution. From Eqs. 13 and 14, itfollows that

�Mt� � �M� exp�K1�R�� �15�

�Rt� � �R� � K1�R��M� exp�K1�R�� �16�

R � ��Rt�3�R��=�Mt� � K1�R� �17�where [Mt] is the total micelle concentration, [Rt] isthe total equivalent concentration of solubilizate, [R]

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is the monomeric solubilizate concentration not solu-bilized into the micelles. Eq. 17 can be rearranged as

��Rt�3�R��=�R� � K1�Mt� �18�where �Mt� � �Ct3�S��=n and Ct is the total surfac-tant concentration.

4.2.3. Molar extinction coe¤cientThe molar extinction coe¤cients (O) were deter-

mined from the optical absorbances of solubilizatesat ¢ve di¡erent concentrations in micellar solutions.The slope of the absorbance plotted against the sol-ubilizate concentrations was used for the extinctioncoe¤cient determination. For polycyclic aromaticcompounds, the wavelengths of the absorption peaksabove the CMC were di¡erent from those below theCMC, which indicates that the dielectric environ-ments surrounding these solubilizates are di¡erentbelow and above the CMC. This observation sup-ports the argument that these solubilizates are solu-bilized into NaC micelles above the CMC. The Ovalues in pure water were determined by the samemethod as described above. The O values used forthe present solubilizates are summarized in Table 1.

4.2.4. Solubilizate concentrationThe slight increase in concentration for anthracene

and pyrene below the CMC is due to the presence ofweak interactions between the solubilizate monomerand cholate monomer. The formation of micellescauses solubilizate concentration to increase compa-ratively rapidly above the CMC. This can be seen inthe CMC value ranges from 12 to 15 mM judgingfrom the data in micellization experiment and Figs. 8and 9. The measured CMC value agrees with thereference data [1,17,37].

The ¢nal solubilizate concentration in the cholatesolutions for alkylbenzenes depends on the theiramount placed inside the apparatus. Those for thearomatic compounds are the maximum additive con-centration, which is a saturation concentration in thepresence of a coexisting solid phase. Therefore, thelatter depends on temperature, pressure, and thecholate concentration, as mentioned above.

4.2.5. Thermodynamic parametersWe can determine the ¢rst stepwise association

constant �K1� at di¡erent Ct concentrations or at

Table 2The average micellar aggregation number (n), the average num-ber of solubilizate per micelle (R), the association constant, andthe Gibbs energy change of solubilization

Solubilizate n R K1/mol31

dm3vG0/kJmol31

Benzene 8.6 0.026 6.2U101 310.210 0.033 7.7U101 310.813 0.047 1.1U102 311.616 0.065 1.4U102 312.3

Toluene 8.6 0.12 3.4U102 314.510 0.15 4.2U102 315.013 0.21 5.7U102 315.716 0.25 7.1U102 316.3

Ethylbenzene 8.6 0.20 7.7U102 316.510 0.26 1.0U103 317.213 0.39 1.6U103 318.216 0.51 2.0U103 318.9

n-Propylbenzene 8.6 0.22 2.4U103 319.210 0.28 3.1U103 319.913 0.39 4.8U103 321.016 0.58 6.7U103 321.8

n-Butylbenzene 8.6 0.23 8.1U103 322.310 0.34 1.2U104 323.313 0.55 1.8U104 324.416 0.72 2.5U104 325.1

n-Pentylbenzene 8.6 0.61 2.6U104 325.210 0.84 3.6U104 326.013 1.3 5.4U104 327.016 1.7 7.0U104 327.7

n-Hexylbenzene 8.6 0.44 7.3U104 318.910 0.64 1.1U105 328.813 0.93 1.6U105 329.616 1.2 2.0U105 330.2

Naphthalene 8.6 0.42 2.0U103 318.910 0.56 2.7U103 319.613 0.78 3.7U103 320.416 1.0 4.8U103 321.1

Anthracene 8.6 0.0076 3.6U104 326.010 0.011 5.2U104 326.913 0.016 7.7U104 327.916 0.022 1.1U105 328.7

Pyrene 8.6 0.059 1.5U105 329.610 0.087 2.2U105 330.513 0.14 3.5U105 331.716 0.18 4.7U105 332.4

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di¡erent n values from Eq. 18, because the relationbetween Ct and n, has already been determined. Ta-ble 2 shows the �K1� and R values. The �K1� valuesincrease rather rapidly with the n values, i.e. thelarger the n value becomes, the larger is the amountof solubilizate that can be stabilized by solubiliza-tion.

The R values obtained for all solubilizates aresmall enough to con¢rm that the incorporation ofthe solubilizates can not change the intrinsic proper-ties of the parent micelles and that the assumption isreasonable.

The Gibbs energy change of solubilization can beexpressed as

vG0 � 3RT ln K1 �19�

Here, vG0 is made up of the three contributions:

vG0 � l0MR1

3l0M3l0

R �20�

where l0MR1

is the standard chemical potential ofMR1 at in¢nite dilution and l0

M and l0R are the cor-

responding potentials. The vG0 values are also givenin Table 2. In Figs. 10 and 11, respectively, the vG0

values for solubilization at di¡erent mean aggrega-tion numbers of sodium cholate for n-alkylbenzenesand polycyclic aromatic compounds are plotted

against the number of carbon atoms. In both cases,the value of vG0 decreased linearly with the numberof carbon atoms of the solubilizates, suggesting thatthe solubilizates were stabilized by incorporation intosodium cholate micelles as their hydrophobicity in-creased.

Although all the solubilizates are stabilized bytheir incorporation into the cholate micelles withthe vG0 values decreasing with increasing mean ag-gregation number, all the slopes remain constant:33.0 kJ mol31 for n-alkylbenzenes, and 32.0 kJmol31 for polycyclic aromatic compounds. Thisfact is quite characteristic of solubilization in sodiumcholate micelles. In other words, the Gibbs energychange per methylene group usually decreases inmagnitude with decreasing micellar aggregationnumber for the conventional aliphatic surfactants[44]. However, this appears not to be the case forsodium cholate micelles. From the above values, sixcarbons in an arene molecule or a benzene ring areequivalent to 4.0 methylene groups, as far as theGibbs energy change is concerned. The values ofthe slopes can be compared with those for 1-dodec-anesulfonic acid micelles whose aggregation numberis 66 [38,39]: 32.81 kJ mol31 for n-alkylbenzenesand 31.77 kJ mol31 for polycyclic aromatic com-pounds. The above values strongly suggest that so-

Fig. 11. Gibbs energy change of solubilization for di¡erentmean aggregation numbers (n) versus total number of carbonatom in polycyclic aromatic compounds at 298.2 K: numbersin parentheses are total surfactant concentration in mM.

Fig. 10. Gibbs energy change of solubilization for di¡erentmean aggregation numbers (n) versus carbon number of alkyl-chain in n-alkylbenzenes at 298.2 K: numbers in parenthesesare total surfactant concentration in mM.

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dium cholate micelles with their lower aggregationnumbers are better solubilizers for both alkane andaromatic compounds than 1-dodecanesulfonic acidmicelles, despite the latter having a longer hydropho-bic alkylchain and larger aggregation number. Thedi¡erences between 33.0 and 32.8 kJ mol31 forthe alkylbenzenes, and 32.0 and 31.8 kJ mol31 forthe aromatics are quite noticeable. This strongly in-dicates that solubilization into sodium cholate mi-celle brings about larger stabilization for the solubi-lizates in spite of the small mean aggregationnumber. The increased stabilization suggests that in-ner hydrophobic region made of hydrocarbon backsof the steroid nucleus is more hydrophobic than in-ner micelle made of alkylchains of conventional ali-phatic surfactants.

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