Solubilization of “Guest” Molecules into Polymeric Aggregatessites.psu.edu/nagu/wp-content/uploads/sites/54945/2016/06/2001... · Solubilization of “Guest” Molecules into

Solubilization of ªGuestº Molecules intoPolymeric AggregatesR. Nagarajan*Department of Chemical Engineering, 161 Fenske Laboratory, The Pennsylvania State University, University Park,PA 16802, USA

ABSTRACT WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

The solubilization capacity of block copolymer aggregatesin aqueous solutions for hydrophobic guest molecules isreviewed in this paper. Experimental results on thesolubilization of a number of hydrocarbons and mixturesof hydrocarbons, and hydrophobic molecules with vary-ing degrees of polarity are first summarized. We thendescribe in detail a predictive molecular thermodynamictheory of solubilization formulated in our earlier studies.Explicit analytical expressions are provided for thestandard state free energy change associated withsolubilization of hydrocarbons in aggregates havingspherical, cylindrical and lamellar shapes. Utilizing thesefree energy expressions and using only molecularconstants, the core size, corona thickness, and aggrega-tion number of the polymeric aggregates and the volumefraction of the hydrocarbon solubilized in the core havebeen predicted. The characteristics of aggregates formedfrom diblock and triblock copolymers and their soluteuptake capacity are compared and related to the molecu-lar properties of the guest molecules and the blockcopolymers. Further, theoretical results from a systema-tic study of solubilization of several hydrocarbons byaggregates formed of the family of Pluronic1 triblockcopolymers are discussed. Very interestingly, solubiliza-tion is shown to induce a transition in aggregate shapesfrom spheres to cylinders and then to lamellae. The originof such shape transitions is identified in terms of thedifferent free energy contributions. Finally, a fewapplications of solubilization in block copolymer aggre-

gates are briefly mentioned. Copyright 2001 JohnWiley & Sons, Ltd.

KEYWORDS: guest molecules; selective solubilization;diblock and triblock copolymer micelles; polymericaggregates

INTRODUCTION WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

Block copolymer molecules consisting of hydro-phobic and hydrophilic blocks aggregate in aque-ous solutions, forming structures well-known asmicelles. In these micelles, the hydrophilic blocksconstitute the corona or the shell region while thehydrophobic blocks form the core. This phenom-enon is entirely analogous to the formation ofmicelles by conventional low-molecular-weightsurfactants. The concentration at which themicellesare first detected is known as the critical micelleconcentration (CMC). In the case of block copoly-mers, because of the large size of the hydrophobicblock, the CMC is often too small to be measurablewhen compared to the CMCs of low-molecular-weight surfactant systems. One of the most usefulproperties of the micellar aggregates is their abilityto take up hydrophobic guest molecules which areotherwise only sparingly soluble in water. Themicellar core serves as a compatible micro-envir-onment for the water-insoluble guest molecules,thereby enhancing their solubility in water. Thisphenomenon is referred to as solubilization. Thesolubilization of guest molecules by block copoly-mer micelles holds great potential for the develop-

Copyright 2001 John Wiley & Sons, Ltd.

POLYMERS FOR ADVANCED TECHNOLOGIESPolym. Adv. Technol. 12, 23±43 (2001)

* Correspondence to: R. Nagarajan, Department of ChemicalEngineering, 161 Fenske Laboratory, The Pennsylvania StateUniversity, University Park, PA 16802, USA.

ment of aqueous block copolymer solutions asenvironmentally benign substitutes for organicsolvents currently being used in applications suchas reaction media in chemical process industry, aschemical extractants in separation processes, and asindustrial cleaning agents. Other potential applica-tions such as drug delivery are also based on theexploitation of the solubilization process.

Extensive studies of solubilization in aqueoussolutions of conventional surfactants have ap-peared in the literature [1]. In contrast, there aremuch fewer studies of solubilization of hydropho-bic substances in block copolymer micelles. Experi-mental results on the solubilization of aliphatic andaromatic hydrocarbons in micelles formed ofpoly(ethylene oxide)-poly(propylene oxide) (PEO-PPO) and poly(vinyl pyrrolidone)-polystyrene(PVP-PS) block copolymers were presented byNagarajan et al. [2] In this work, the selectivity inthe solubilization was also explored via measure-ments on binary mixtures of hydrocarbons. Slocumet al. [3] presented experimental results on thesolubility of a number of alcohols, ethyl esters,ketones and aldehydes, all of which are typicalcomponents present in food flavors, in PEO-PPOblock copolymer micelles. They also determinedthe selectivity in solubilization of components oforange oil in block copolymer micelles [4]. Hurterand Hatton [5] have conducted a study of solu-bilization of three polycyclic aromatics in solutionsof PEO-PPO-PEO triblock copolymers and four-armed PEO-PPO star block copolymers of varyingmolecular weights and compositions. The locationof solubilized benzene in micelles formed of theabove copolymers were identified by Nivaggioli etal. [6] as the PPO core region of the aggregates. Thesolubilization of xylene in PEO-PPO-PEO triblockcopolymer micelles and the changes in aggregationnumber induced by the solubilizate were measuredby Chu and coworkers [7, 8]. They observed agrowth in the aggregation number and the hydro-dynamic radius of the micelle on solubilization andalso concluded that xylene is present in the PPOcore of the micelle. The solubilization of fluorescentprobes in PEO-PPO-PEO triblock copolymer solu-tions has been investigated by Kabanov et al. [9].

From a theoretical point of view, the solubiliza-tion in PEO-PPO diblock copolymer micelles andPEO-PPO-PEO triblock copolymer micelles weretreated using a mean-field approach by Nagarajanand Ganesh [10, 11] considering spherical as well asnon-spherical shapes for the micellar aggregates.They formulated [12] also an alternate treatmentbased on the star polymer model using the scalingapproach and focusing on spherical micelles.Similar scaling analysis of solubilization in diblockcopolymer aggregates applicable to spherical,cylindrical and lamellar structures has been pre-sented by Dan and Tirrell [13]. The solubilizationphenomenon has been treated using a self-consis-tent mean-field approach taking into accountcomposition inhomogeneities inside the aggregatestructure by Cogan et al. [14], Hurter et al. [15, 16],Linse [17] and Leermakers et al. [18]. The thermo-

dynamics of solubilizate uptake has been treated byLebens and Keurentjes [19] in terms of the enthalpyand the entropy of interactions between thesolubilizate and the micellar core block, the energyof creation of the interface and the two conforma-tions proposed for the poly(ethylene oxide) andpoly(propylene oxide) chains [20].

This review emphasizes our own work for tworeasons. First, experimental results have beenobtained for a number of molecules by system-atically varying the properties of guest molecules.Second, our theoretical treatment of solubilizationis relatively simple. The main advantage of thetheory is that it allows the prediction of all micro-structural features of micelles containing solubili-zates with only minor computational effort.Further, all free energy contributions are given asexplicit analytical functions directly linked tophysicochemical changes accompanying solubiliza-tion, and involving only molecular constants andgeometrical variables. Such an explicit link betweenthe free energy of solubilization on the onehand andthemolecular properties of the guest molecules andthe block copolymer on the other hand, allows amolecular interpretation of the uptake of guestmolecules by the block copolymer aggregates.

This paper is organized as follows. In the nextsection, we discuss experimental results on thesolubilization capacity of block copolymer aggre-gates for many guest molecules. This is followed bya thermodynamic treatment of solubilization inblock copolymer micelles using the mean-fieldapproach in the section on Molecular Theory ofSolubilization. The free energy expressions arepresented in a general manner applicable to bothdiblock and triblock copolymers and also tospherical, rod-like and lamellar aggregates. TheSolubilization Ð Predictions of Molecular Theorysection summarizes the main results, with empha-sis on the importance of the properties of thesolubilizates, comparison between the diblock andtriblock copolymers, and a systematic investigationof structural transitions in Pluronic triblock co-polymers induced by various solubilizates. The lastsection refers to yet other types of guest moleculessolubilized in block copolymer aggregates andtheir importance to practical applications.

CAPACITY OF AGGREGATES FORGUEST MOLECULES WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

The solute uptake capacity of block copolymermicelles has been represented in the literature inmany forms. These include, micelle±water partitioncoefficients (defined as the ratio between theconcentration of the solute inside the micelle andthe concentration of the solute that is molecularlydispersed in the aqueous phase, the concentrationsbeing in mole fractions or molarities), the limitingamount of solubilization (which occurs when theaqueous solution of micelles coexists with an excesssolubilizate phase), volume fraction in the micellecore and the molar solubilization ratio, MSR (which

Copyright 2001 John Wiley & Sons, Ltd. Polym. Adv. Technol., 12, 23±43 (2001)

24 / Nagarajan

is the ratio of the moles of guest molecules to themoles of polymer molecules in the aggregate).

Solubilization of Hydrocarbons

The solubilization capacity of block copolymermicelles for aromatic and aliphatic hydrocarbonshave been determined using the gas chromato-graphic method [2]. Two block copolymer mol-ecules, PEO-PPO having a molecular weight of12,500 and containing 30 wt% PPO (correspondsapproximately to 200 ethylene oxide and 64propylene oxide segments) and PVP-PS of unspe-cified molecular weight having 60 wt% PS wereused in these experiments. The copolymers werenot well characterized with respect to their poly-dispersity and there is some uncertainty as towhether the PEO-PPO copolymer is a diblock ortriblock copolymer [5]. The extent of solubilizationof the guest molecules in aqueous solutions of 10wt% PEO-PPO block copolymer or 20 wt% PVP-PSblock copolymer are listed in Table 1. Also listed forcomparison are the results obtained in solutions ofa conventional low-molecular-weight surfactant,sodium dodecyl sulfate (SDS). The limitingamounts solubilized are expressed as mmolessolubilized per gram of the hydrophobic block ofthe copolymer (or of the hydrophobic surfactanttail). Also listed are some important molecularcharacteristics of the solubilizates including theirmolecular volume (vJ), their interfacial tensionagainst water (�JW), and their Hildebrand±Scatch-ard solubility parameter (�J) values. The subscript Jis used to denote the solubilizate while W refers tothe solvent water. Subscripts A and B denote thehydrophobic and hydrophilic blocks of the copo-lymer, respectively.

The most notable feature of the results is thelarge difference in the uptake capacity for aromatichydrocarbons compared to aliphatic hydrocarbons.The four-fold difference between the solubilizedmoles of benzene and hexane in the low-molecular-weight surfactant SDS is replaced by a 17-folddifference in the PEO-PPO block copolymer and bya 40-fold difference in the PVP-PS block copolymer.

The amounts solubilized in conventionalsurfactant micelles have been correlated [21]with sufficient accuracy to the molecular volumeof the solubilizates (vJ), and also to a non-di-mensional volume-polarity parameter defined as��JWv2=3J =kT�where k is the Boltzmann constant andT is the absolute temperature. In defining thevolume-polarity parameter, the solubilizate±waterinterfacial tension is taken as a measure of thesolubilizate's polarity and the consequent inter-facial activity. The correlations show that theamount solubilized for a homologous family ofsolubilizates decreases with increasing molecularsize of the solubilizate. The difference between thearomatic and aliphatic solubilizates of similarmolecular volumes (toluene and cyclohexane, forexample) has been explained in terms of thepolarity and the consequent interfacial activity ofthe aromatics and the correlation based on thevolume-polarity parameter has been found to bequite satisfactory for many surfactants. [21]

In the case of polymeric aggregates, one canexpect the interactions between the core block A ofthe copolymer and the solubilizate J to influence thesolubilization capacity. These interactions arerepresented by the Flory interaction parameter wAJ

which can be estimated [2] from knowledge of theHildebrand solubility parameters of both A and Jvia the relation �AJ = (�Aÿ �J)2 vJ/kT, where �A isthe solubility parameter for the core block. Taking�A to be 19 MPa1/2 for PPO, 18.6 MPa1/2 for PS, and15.94 MPa1/2 for the dodecyl tail of the SDS, andusing the molecular properties of solubilizateslisted in Table 1, we can calculate the Floryinteraction parameters for all the solubilizates.The measured solubilization capacity has beencorrelated with the Flory interaction parameter inthe form

MSR � a �ÿbAJ � a��A ÿ �J�2vJ

kT

" #ÿb�1�

where the positive constants a and b are dependenton the block copolymer molecule. One can see that

TABLE 1. Solubilization Capacity of Block Copolymer Micelles for Aromatic and Aliphatic Hydrocarbons at 25°C andSome Molecular Properties of the Solubilizates

Solubilizate

Molecular properties of solubilizatesmmoles solubilized per gram of

hydrophobic block

vJ (AÊ 3) �JW (dyn/cm) �J (MPa1/2) PEO-PPO PVP-PS SDS

Benzene 146 33.93 18.80 11.67 30 9.9Toluene 176 36.1 18.19 6.33 14.8 8o-Xylene 200 36.1 18.40 4.0 31.6 4.43Ethyl benzene 204 38.4 17.99 5.67 26.7 ÐCyclohexane 179 50.2 16.76 1.97 3 4.6Hexane 217 50.7 14.92 0.667 0.77 2.39Heptane 243 51.2 15.13 0.567 0.47 ÐOctane 270 51.5 15.53 0.5 0.18 ÐDecane 323 52 15.74 0.387 0.072 1.18


Solubilization of Guest Molecules / 25

the aromatic solubilizates with the small wAJ valuesare good solvents for the PPO and the PS blocksand give rise to large MSRs. In contrast, thealiphatic solubilizates are poor solvents for theseblocks and display low values for the MSR. Theabove correlation has been used to describe thesolubilization data in also PS-poly(methacrylicacid) (PS-PMA) block copolymer micelles obtainedfor 18 different solubilizates including aliphatic,cyclic and aromatic hydrocarbons, chlorinatedhydrocarbons and esters. [22] In this work, thePS-solubilizate interaction parameters have notbeen calculated from the solubility parameters buthave been obtained experimentally using inversegas chromatography.

The partition equilibrium constants K fortoluene, naphthalene and phenanthrene in PVP-PS block copolymer solutions reported in theliterature, [23] are in accord with the first publishedreport [2] of the large solubilization capacity ofPVP-PS block copolymer micelles for aromatics.The partition coefficient of naphthalene solubilizedin various PEO-PPO-PEO triblock copolymer mi-celles [24] was found independent of the blockcopolymer concentration. Its variation with themolecular weight and the composition of the blockcopolymer suggests that changes in the aggregatesize influence the partition coefficient. Solubiliza-tion isotherms of toluene, benzene, chlorobenzeneand p-xylene have been obtained in a number ofPEO-PPO-PEO triblock copolymer micelles byGadelle et al. [25]. The partition coefficient, K, isfound to be strongly dependent on the amount ofsolubilizate present in the micelles. At very lowsolubilizate concentrations, an increase in K withincreasing amount of solubilizate is seen for low-molecular-weight block copolymers suggestingthat the presence of the solubilizate is favoringthe aggregate formation. At higher solubilizate

concentrations, K decreases with increasingamount of solubilizates similar to the behaviorshown by small surfactant molecules. Further, thepartition coefficients are found to depend on theblock copolymer concentration for the lower mol-ecular weight block copolymers and concentration-independent for higher molecular weight blockcopolymers.

Studies on the solubilization of a drug moleculeestriol in a PEO-PPO-PEO triblock copolymershowed [26] that the solubilization is driven byfavorable entropy changes as may be expected for avery hydrophobic solubilizate. The estimated parti-tion coefficient shows an increase in value beyond atransition concentration of the block copolymer; thetransition has been attributed to a possible growthin the aggregate. The increase in K with increasingtemperature has been attributed to a possibledecrease in wAJ and also to an increase in theaggregate size. Similar increases in K have beenreported [9] for hydrophobic fluorescent probemolecules in PEO-PPO-PEO triblock copolymermicelles.

Solubilization of Alcohols, Ketones and Esters

Oxygenated compounds such as alcohols, esters,ketones, aldehydes, phenols, ethers, etc., are pre-sent in essential oils and are important as flavorcompounds in food and beverage industries. Thesolubilization of these compounds in micellarsolutions offers a way of preparing flavor formula-tions that are thermodynamically stable. Slocum etal. [3, 4] have measured the solubilization capacityof PEO-PPO block copolymer micelles for 2-ketones, 1-alcohols and ethyl esters by the turbiditymethod in solutions containing 3 wt% PEO-PPO(30% PPO, MW = 12,500) and the results aresummarized in Table 2. Also listed are the aqueous

TABLE 2. Solubilization Capacity of PEO-PPO (30% PPO, MW = 12,500) Block Copolymer Micelles for Ketones, Alcoholsand Ethyl Esters at 25°C

Solubilizate J vJ (AÊ 3)Solubility in water(mole fraction)

mmoles solubilizedper gram of PPO

2-Heptanone 232.6 6.82� 10ÿ4 0.4182-Octanone 259.7 1.6� 10ÿ4 0.6042-Nonanone 286.8 4.74� 10ÿ5 0.8772-Undecanone 340.9 2.39� 10ÿ6 0.571-Octanol 263.0 8.13� 10ÿ5 1.7331-Nonanol 290.1 1.78� 10ÿ5 1.5031-Decanol 317.1 4.17� 10ÿ6 1.0891-Undecanol 344.2 1.14� 10ÿ6 0.572Ethyl propionate 191.3 2.67� 10ÿ3 7.974Ethyl pentanoate 245.4 3.19� 10ÿ4 0.517Ethyl hexanoate 272.4 8.0� 10ÿ5 0.369Ethyl octanoate 326.5 7.4� 10ÿ6 0.27Ethyl nonanoate 353.5 2.87� 10ÿ6 0.495Ethyl undecanoate 407.7 3.88� 10ÿ7 0.152


26 / Nagarajan

solubility for these molecules. [27] Unlike thearomatic and aliphatic hydrocarbons, the oxyge-nated molecules are capable of interacting via notonly dispersion interactions, but also polar andhydrogen bonding interactions. The measurementsshow that the solubilization capacity of the 1-alcohols decreases with increasing molecular sizeas was observed for the hydrocarbon families.Ethyl esters show a similar trend but the observedsolubilization capacity for ethyl nonanoate is muchlarger than that expected based on the molecularsize dependence. For 2-ketones, the solubilizationcapacities do not show a monotonic decrease withincreasing molecular size of the solubilizates. Theseresults clearly suggest that in addition to molecularsize effects, there must be competing interactionaleffects also present leading to non-monotonicsolute uptake behavior.

Solubilization of Hydrocarbon Mixtures

The single component solubilization results inTable 1 suggest that solubilization will be selectivefor aromatic molecules when mixtures of aromaticand aliphatic molecules are solubilized. This hasbeen demonstrated by experimental results [2]obtained in the PEO-PPO and PVP-PS blockcopolymer systems for binary mixtures of benzeneand hexane. To illustrate the results, the amounts ofeach hydrocarbon solubilized as a function of thecomposition of the bulk solubilizate phase thatcoexists with the micellar solution are shown in Fig.

1 for the PVP-PS block copolymer. One can observethat over the entire composition range, the amountof hexane solubilized is very small. Such selectivityfor benzene over hexane was seen also in the case oflow-molecular-weight surfactants such as SDS.However, in SDSmicelles the selectivity diminisheswhen the size of the aromatic solubilizate increasesfrom benzene to xylene as shown by the data inTable 1. In contrast, the block copolymer systemsdisplay large selectivity for aromatic hydrocarbonsof differing sizes as can be seen from the results forthe PVP-PS copolymer in Table 1. In this case, thecompatibility of the various aromatic solubilizateswith the PS block of the micellar core results in highselectivity for these molecules irrespective of theirmolecular size variations. Such selectivity in thesolubilization behavior is of practical importance tochemical separations.

MOLECULAR THEORY OFSOLUBILIZATION WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

Experimental studies suggest that the measureduptake capacity for the guest molecules is depen-dent on their molecular sizes, their interactionswith the core forming block, and the size andcomposition of the block copolymer. A quantitativetheory that can explain these observations has beendeveloped by Nagarajan and Ganesh [10, 11, 28, 29]later. The model has been used to predict thesolubilization capacity of many block copolymerswith PEO as the hydrophilic block and PPO as thehydrophobic block. Therefore, in the following text,we identify the two polymer blocks to be PEO andPPO, although the calculations can be performedfor any other type of block copolymer as well.

Size and Composition Distribution of Aggregates

One starts by considering the aqueous solutionwhich is made up of solvent molecules, singlydispersed copolymer and solubilizate moleculesand micelles of various sizes and compositions (orequivalently, various aggregation numbers andvolume fractions of solubilizates). Each of thespecies in the solution, including micelles ofdifferent sizes and compositions, is treated as adistinct chemical component. The size and compo-sition distribution of micelles at equilibrium isobtained by minimizing the total free energy of thesystem. In writing the system free energy, thestandard state of the solvent is defined as the puresolvent whereas the standard states of all the othercomponents are taken as those at infinitely dilutesolution conditions. The standard chemical poten-tials of the solvent water (W), the singly dispersedcopolymer, the singly dispersed solubilizate (J) andmicelles of aggregation number g containing jsolubilizate molecules, are denoted by �oW; �

o1; �

o1J

and �og , respectively. (Note that g and j refer tothe total numbers of molecules for spherical aggre-gates, numbers per unit length in the case of cylin-drical aggregates and numbers per unit area in thecase of lamellae.) Denoting the mole fraction of

FIGURE 1. Amounts of benzene and hexane solubilized inthe aqueous PVP-PS micellar solution as a function of thecomposition of the hydrocarbon phase coexisting with it.



species i by Xi, the micelle size and compositiondistribution equation can be written in the form:

Xg � Xg1X

j1J expÿ

�og ÿ g�o1 ÿ j�o1JkT

� ��2�

In writing eq. (2), it is assumed that eitherintermicelle interactions are not present or thatthey do not affect the size distribution. Also thesystem entropy of the multicomponent solution iswritten as for an ideal solution. In dilute solutionssuch as those of interest here, the intermicellarinteractions are not important. Therefore, one mayneglect the free energy contributions associatedwith such interactions. The consequences of using afew plausible models of system entropy in thetheory of micellization have been analyzed in detailby Nagarajan [30] with the conclusion that thestructural features of aggregates are unaffected bythe choice of the models.

If one wants to calculate the saturation amountof solubilization that is possible inside the micelle,then the concentration of the singly dispersedsolubilizate X1J in water should be its saturationconcentration X�1J. This condition is defined by theequilibrium relation

�HJ � �o1J � kT ln X�1J �3�

Since an excess solubilizate phase comes intoexistence when the aqueous solution is saturatedwith the solubilizate, the standard state �HJ of thesolubilizate refers to a pure solubilizate phase.Denoting by f the fractional saturation of waterwith the solubilizate (i.e. X1J=X�1J), the micelle sizeand composition distribution equation becomes:

Xg � Xg1f

j=g expÿ g��ogkT

� �;

where ��og ��ogg ÿ �o1 ÿ

jg�

HJ

� �The factor ��ogrepresents the change in the stan-dard state free energy when a singly dispersedblock copolymer molecule in water and j/g solubi-lizate molecules in their pure phase are transferredto an isolated micelle in water. The solubilizationlimit is achieved for f = 1, when the aqueousmicellar phase coexists with the pure solubilizatephase. All predictions given in this paper are forthis condition.

Pseudophase Treatment of Solubilization

In order to reduce the numerical computationalefforts, the micelle containing the solubilizate canbe represented as a pseudophase in equilibriumwith the singly dispersed solubilizate and copoly-mer molecules in solution. For aggregates exhibit-ing narrow size and composition distribution, thisrepresentation provides results practically identicalto those obtained from the detailed size distributioncalculations. The equilibrium characteristics of the

micelle in the pseudophase approximation areobtainable from the condition:

@

@g

��og

kT

� �� 0;

@

@j

��og

kT

� �� 0 at g � gopt; j � jopt

�5�where gopt and jopt refer to the numbers of blockcopolymer and solubilizate molecules, respec-tively, constituting the optimal or equilibriumaggregate. The cmc in the pseudophase approxi-mation is calculated from:

XCMC � exp��og

kT

� �at g � gopt; j � jopt �6�

The magnitude of ��og controls the CMC as shownby eq. (6). In contrast, the equilibrium structuralfeatures of the micelle are determined by how thisstandard free energy difference depends on thevariables g and j as is evident from eq. (5). Anexpression for this free energy difference dependson the geometrical features of the micelle, whichare specified later.

Structure of Aggregates

Two structural descriptions of aggregates can bevisualized as shown in Fig. 2, depending upon howthe solubilizate is contained inside the blockcopolymer micelle. These two descriptions resultby analogy with the structural models employedfor simple solubilization and microemulsification,respectively, in systems involving low-molecular-weight surfactants. The type (a) structure isanalogous to that used for simple solubilization.Here, the micellar core is made up of the solventincompatible A blocks and the solubilizate J. Thesolvent compatible B blocks and solvent W arepresent in the corona region of the micelle. The type(b) structure is analogous to that used for dropletmicroemulsions, with the core region separatedinto two parts. Pure solubilizate J is allowed to existin the inner core of the micelle. This solubilizatedomain is surrounded by the outer core regionconsisting of the A block and the solubilizate J. Thecorona of the micelle contains the B block and thesolvent W as in the type (a) structure. In the absenceof a pool of pure solubilizate J in the inner core, thetype (b) structure reduces identically to type (a).Free energy calculations of the kind describedbelow showed for all block copolymers examined,the condition of minimum free energy alwaysoccurred corresponding to a zero size for the puresolubilizate pool. Thus, the thermodynamic equili-brium criterion always favored the occurrence ofthe type (a) structure. Consequently, free energyexpressions corresponding to only the type (a)structure are discussed here. Although Fig. 2depicts spherical aggregates only, analogous struc-tures can be visualized having cylindrical andlamellar shapes.

(4)


28 / Nagarajan

Geometrical Relations for Aggregates

The shape of the aggregate and the assumption ofincompressibility lead to the geometrical relationssummarized in Table 3 for different morphologies[28]. We denote themolecular volumes of the A andthe B segments, the solubilizate and the solvent byvA, vB, vJ and vW, respectively. The characteristiclengths of the A and the B segments are denoted byLA�� v1=3A � and LB�� v1=3B �. The variables NA and NB

refer to the number of segments of block A and

block B for the AB diblock as well as the BABtriblock copolymers, implying that the BAB triblockcopolymer has two terminal blocks of size NB/2attached to a middle block of size NA. We use thevariable R to denote the hydrophobic core dimen-sion (radius for sphere or cylinder and half bilayerthickness for lamella), D for the corona thickness,and s for the surface area of the aggregate core perconstituent block copolymer molecule. The num-bers of molecules g and j, the micelle core volumeVC, and the corona volume VS all refer to the totalquantities in the case of spherical aggregates,quantities per unit length in the case of cylindricalaggregates and quantities per unit area in the caseof lamellae. The core volume VC is calculated as thesum of the volumes of the A blocks and thesolubilizate molecules, VC = g NA vA� j vJ. Thevolume fraction of the solubilizate molecules in thecore is denoted by � (= j vJ/(g NA vA� jvj)). Theconcentrations of segments are assumed to beuniform in the core as well as in the corona, with'A standing for the volume fraction of the Asegments in the core ('A = 1ÿ �), and 'B for thevolume fraction of the B segments in the corona. Ifany three structural variables are specified all theremaining geometrical variables can be calculatedthrough the relations given in Table 3. Forconvenience, R, D and � (or 'A) are chosen as theindependent variables.

Model for Free Energy of Solubilization

An expression for the free energy of solubilization��ogdefined in eq. (4) is formulated by visualizingall the physicochemical changes accompanying thetransfer of a singly dispersed copolymer moleculefrom the infinitely dilute aqueous solution stateand the solubilizate molecule from its pure phase toan isolated micelle in the infinitely dilute solutionstate. Firstly, the transfer of the solubilizate and thesingly dispersed copolymer to the micellar core isassociated with changes in the state of dilution andin the state of deformation of the A block, includingthe swelling of the A blocks inside the core by thesolubilizate J. Secondly, the B block of the singlydispersed copolymer is transferred to the coronaregion of the micelle and this transfer process alsoinvolves changes in the states of dilution anddeformation of the B block. Thirdly, the formationof the micelle localizes the copolymer such that theA block is confined to the core while the B block isconfined to the corona. Fourthly, the formation of

FIGURE 2. Schematic representation of a sphericalmicelle containing the solubilizate. The darker lines denotethe hydrophobic block and the lighter lines, the hydrophilicblock. In (a) all of the solubilizate molecules interact with thecore block. In (b), a part of the solubilizate molecules arepresent in a separate domain while the remaining interactwith the core block.

TABLE 3. Geometrical Properties of Spherical, Cylindrical and Lamellar Aggregates

Property Sphere Cylinder Lamella

VC 4pR3/3 pR2 2RVS VC [(1� D/R)3ÿ 1] Vc [(1� D/R)2ÿ 1] Vc [(1� D/R)ÿ 1]g Vc ('A/vA) Vc ('A/vA) Vc ('A/vA)s 3 vA/ (R'A) 2 vA/ (R'A) vA/ (R'A)'B (vB/vA) 'A (VC/VS) (vB/vA) 'A(Vc/Vs) (vB/vA) 'A(Vc/Vs)



the micelle is associated with the generation of aninterface between the micelle core made up of Ablocks and solubilizate J and the micelle coronaconsisting of solvent W and B blocks. All thesechanges contribute to the free energy of solubiliza-tion in the case of both diblock and triblockcopolymers. Further, in the case of a BAB triblockcopolymer, folding or loop formation of the A blockoccurs ensuring that the B blocks at the two endsare in the aqueous domain while the folded A blockis within the hydrophobic core of the micelle. Thisprovides an additional free energy contribution.The overall free energy of solubilization can beobtained as the sum of the above individualcontributions:

��og� � ��og�A;dil � ��og�A;def� ��og�B;dil � ��og�B;def� ��og�loc � ��og�int � ��og�loop �7�

Expressions for each of these contributions areformulated below.

Change in State of Dilution of Block A. In the singlydispersed state of the copolymer molecule in water,the A block is in a collapsed state minimizing itsinteractions with the solvent. The region consistingof the collapsed A block with some solvententrapped in it is viewed as a spherical globule,whose diameter 2R?A is equal to the end-to-enddistance of block A in the solvent. The volume ofthis spherical region is denoted by V?A. The chainexpansion parameter aA describes the swelling ofthe polymer block A by the solvent W.

V1A � 4�R31A3 ; 2R1A � sAN

1=2A LA;

�A � 6�

� �1=3Nÿ1=6A '

ÿ1=3A1

where 'A1 (=NA vA/V?A) is the volume fraction ofA segments within the monomolecular globule.The ®rst equality in eq. (8) follows from geometry,while the second equality is based on the de®nitionof the chain expansion parameter aA, taking�N1=2

A LA�as the unperturbed end-to-end distanceof block A. The third equality is obtained bycombining the ®rst two in conjunction with thede®nition for 'A1. Applying the suggestion of deGennes [31], the volume fraction 'A1 is calculated[32] from the condition of osmotic equilibriumbetween the monomolecular globule treated as adistinct phase and the solvent surrounding it.

ln �1ÿ 'A1� � 'A1 � xAW'2A1 � 0 �9�

In eq. (9), wAW is the Flory interaction parameterbetween the pure A polymer and water. In themicelle, the A block is con®ned to the core regionwhere it is swollen by the solubilizate J. We

consider this region to be uniform in concentrationwith a volume fraction 'A of A segments and �(=1ÿ 'A) of the solubilizate. A mean-®eld descrip-tion is employed for calculating the free energy ofthis region. The difference in the dilution of block Afrom its singly dispersed state to the micellizedstate makes a free energy contribution given by therelation

��og�A;dilkT

� NAvAvJ

1ÿ 'A

'Aln �1ÿ 'A� � vA

vJ�1ÿ 'A��AJ

� �

ÿNAvAvW

1ÿ 'A1

'A1ln �1ÿ 'A1�

�� vAvW�1ÿ 'A1��AW � �AWL2A

kT

� �6

�AN1=2A

#�10�

In this equation, the ®rst two terms account for theentropic and enthalpic contributions arising fromthe mixing of pure A block and the pure solubili-zate J within the micellar core. They are written inthe form of the Flory expression for the swelling ofa network [33] by a solvent. The third and thefourth terms account for the entropic and enthalpicchanges associated with the removal of A blockfrom its in®nitely dilute state in water to a pure Astate. These terms are written in the framework ofthe Flory expression [33] for an isolated polymermolecule. The last term accounts for the fact thatthe interface of the globule of the singly dispersedA block disappears on micellization. This term iswritten as the product of the interfacial tension(�AW) between pure A and solvent W, the surfacearea of the globule �4�R2

1A�and the factor 'A1

(volume fraction of the polymer A in the globule) toaccount for the reduction in the contact areabetween the block A and solvent W caused by thepresence of some water molecules inside themonomolecular globule. If the interfacial tension�AW is not available from direct measurements, itcan be estimated using the relation �AW = (�AW/6)1/2 (kT/L2), where L � v1=3W .Such a relation isusually employed for the calculation of polymer±polymer interfacial tensions.

Change in State of Deformation of Block A. In the singlydispersed state of the copolymer, the conformationof the A block is characterized by the chainexpansion parameter aA which is the ratio betweenthe actual end-to-end distance and the unperturbedend-to-end distance of the polymer block. The freeenergy of this deformation is written using theFlory expression [33] derived for an isolatedpolymer molecule. Within the micelle, the A blockis stretched non-uniformly, with the chain endsoccupying a distribution of positions within thecore while ensuring that the core has an uniformconcentration. The free energy contribution allow-ing for non-uniform chain deformation is calcu-

(8)


30 / Nagarajan

lated using the analysis of chain packing pioneeredby Semenov [34]. In the case of a BAB triblockcopolymer, the A block deformation is calculatedby considering the folded A block of size NA to beequivalent to two A blocks of size NA/2. On thisbasis, one obtains:

��og�A;defkT

� qp�2

80

� �R2

�NA=q�L2A

� �ÿ 3

2��2

A ÿ 1� ÿ ln �3A

� � �11�

where q = 1 for a AB diblock copolymer and q = 2for a BAB triblock copolymer having a middlehydrophobic block. The parameter p is dependenton aggregate shape and has the value of 3 forspherical micelles, 5 for cylinders and 10 forlamellae [34, 35]. In eq. (11), the ®rst term repre-sents the A block deformation free energy in themicelle while the second term corresponds to thedeformation free energy in the singly dispersedcopolymer.

Change in State of Dilution of Block B. In the singlydispersed state of the copolymer, the polymer blockB is swollen with the solvent. As mentioned before,NB denotes the size of the B block for the AB diblockcopolymer while for a symmetric BAB triblockcopolymer, the end blocks are of equal size NB/2.We consider this swollen B block to be a sphere,whose diameter 2R?B is equal to the end-to-enddistance of isolated block B in the solvent. Thevolume of this spherical region is denoted by V?B

while 'B1 (=NB vB/V?B) is the volume fraction of Bsegments within the monomolecular globule.

V1B � 4�R31B

3; 2R1B � �B�NB=q�1=2LB �12�

The second equality in eq. (12) is based on thede®nition of the chain expansion parameter aB,which can be estimated using the expressiondeveloped by Flory [33]. In the Flory expressionfor aB, Stockmayer [36] has suggested decreasingthe numerical coef®cient by approximately a factorof two to ensure consistency with the resultsobtained from perturbation theories of excludedvolume. Consequently, one can estimate aB as thesolution of:

�5B ÿ �3

B � 0:88�1=2ÿ �BW��NB=q�1=2 �13�

where wBW is the Flory interaction parameterbetween the B block and water.

In the micelle, the B blocks are present in thecorona region of volumeVS. This region is assumedto be uniform in concentration with 'B (=gNB vB/VS) being the volume fraction of the B segments inthe corona. The free energy of the corona region canbe written using the Flory expression [33] for anetwork swollen by the solvent. Therefore, thedifference in the states of dilution of the B block on

micellization provides the following free energycontribution:

��og�B;dilkT

� NBvBvW

1ÿ 'B

'Bln �1ÿ 'B� � vB

vW�1ÿ 'B��BW

� �

ÿNBvBvW

1ÿ 'B1

'B1ln �1ÿ 'B1� � vB

vW�1ÿ 'B1��BW

� ��14�

The first two terms in eq. (14) describe the entropicand enthalpic contributions to the free energy ofswelling of the B block by the solvent in the coronaregion of the micelle while the last two terms referto the corresponding contributions in the singlydispersed copolymer molecule.

Change in State of Deformation of Block B. In the singlydispersed state, the B block has a chain conforma-tion characterized by the chain expansion para-meter aB. In the micelle, the B block is stretchednon-uniformly over the micelle corona so as toensure that the concentration in the corona region isuniform. Semenov [34] has shown that the estimatefor the chain deformation energy assuming that thetermini of all B blocks lie at the distance D from thecore surface is not very different from thatcalculated assuming a distribution of chain terminiat various positions within the corona. On thisbasis, one can write [28, 29]:

��og�B;defkT

� q3

2

LBR

�s=q�'BP

� �ÿ q

3

2��2

B ÿ 1� ÿ ln �3B

� ��15�

where s is the surface area per molecule of themicelle core, q = 1 for AB diblock and 2 for BABtriblock, as mentioned before, and P is a shape-dependent function given by P = (D/R)/[1� (D/R)] for spheres, P = ln [1� (D/R)] for cylinders andP = (D/R) for lamellae [28, 29]. The ®rst term in eq.(15) represents the free energy of deformation ofthe B block in the micellar corona while the secondterm denotes the corresponding free energy in thesingly dispersed copolymer molecule.

Localization of the Copolymer Molecule. On micelliza-tion, the copolymer becomes localized in the sensethat the joint linking blocks A and B in thecopolymer is constrained to remain in the inter-facial region rather than occupying all the positionsavailable in the entire volume of the micelle. Theentropic reduction associated with localization ismodeled using the concept of con®gurationalvolume restriction. Thus, the localization freeenergy is calculated on the basis of the ratiobetween the volume available to the A±B joint inthe interfacial shell of the micelle (surrounding thecore and having a thickness LB) and the total



volume of the micelle.

��og�lockT

� ÿq ln dLB

R�1�D=R�d" #

�16�

Here, d refers to the dimensionality of aggregategrowth and is 3 for spherical micelles, 2 forcylinders and 1 for lamellae [28, 29].

Formation of Micellar Core±Solvent Interface. Whenmicelle forms, an interface is generated between thecore region consisting of the A block and thesolubilizate J and the corona region consisting ofthe B block and the solvent W. The free energy offormation of this interface is estimated as theproduct of the surface area of the micellar coreand an interfacial tension characteristic of thisinterface. The appropriate interfacial tension is thatbetween a solution of block A and solubilizate J inthe micelle core and a solution of block B andsolvent W in the micellar corona. Since the coronaregion is often very dilute in block B, the interfacialtension can be approximated as that between thesolvent W and a solution of the A block and thesolubilizate J in the micelle core. Denoting thepolymer A±solvent W interfacial tension by �AW

and the solubilizate J±solvent W interfacial tensionby �JW, the free energy of generation of the micellarcore±solvent interface is calculated from:

��og�intkT

� �aggkT

s; �agg � �AW'A � �JW�1ÿ 'A��17�

Here, the interfacial tension of a polymer solutionof block A and solubilizate J against anotherliquidW is approximated to be the composition-averaged interfacial tensions of pure polymer Aand pure solubilizate J against the solvent W. Thevolume fraction is used as the composition vari-able. Such a simple dependence of the interfacialtension on bulk solution composition is notgenerally obeyed in case of free solutions ofpolymers or of low-molecular-weight components.The origin of the deviation from linearity lies in thepreferential adsorption or depletion of one of thecomponents at the interface, which causes thesurface composition to differ from the bulkcomposition [37]. However, the micellar interfaceis different from the interface of a free polymersolution. Speci®cally, because of the localization ofthe A±B link at the interface, the segments of the Ablocks are forced to be at the interface independentof any selective adsorption or depletion. Thiswould diminish somewhat the difference betweenthe surface and bulk compositions in the micellarcore when compared to that in a free polymersolution. Consequently, the composition-averagingof interfacial tension expressed by eq. (17) is takenas a reasonable approximation in the presentcalculations. An alternate approach to estimatingthe interfacial tension by calculating the interface

composition between two bulk solutions has beenexplored in our study of solubilization in low-molecular-weight surfactant micelles [35].

Backfolding in Triblock Copolymer. The backfolding ofthe middle block in a BAB triblock copolymercontributes an entropic term to the free energy ofsolubilization. This contribution is absent for thecase of a diblock copolymer. Jacobson and Stock-mayer [38] showed that the reduction in entropy forthe condition that the ends of a linear chain of Nsegments are to lie in the same plane or on one sideof a plane is proportional to lnN. Therefore, theassumption that the backfolding of the middleblock in the micelle follows the same functionalform is made. Hence, the backfolding entropymakes the following contribution in the case of aBAB copolymer.

��og�loopkT

� 3

2� ln �NA� �18�

Here, b is an excluded volume parameter which isequal to unity when the excluded volume effectsare negligible and larger than unity when theseeffects become important. In our calculations, b istaken to be unity [28, 29].

Molecular Constants

To perform quantitative calculations, the values ofmolecular constants appearing in eqs (8)±(18) areneeded. The molecular volumes of the solubili-zates, their solubility parameters, and the inter-facial tensions between the solvent and thesolubilizates are all listed in Table 1. Values forthe Flory±Huggins interaction parameters wAJ havebeen estimated utilizing the solubility parameters.Illustrative calculations have been carried out forPEO-PPO diblock (EXPY) and PEO-PPO-PEO tri-block (EXPYEX) copolymers. The molecular vol-umes of the repeating units are taken to be 96.5 AÊ 3

for propylene oxide and 64.6 AÊ 3 for ethylene oxidewhile the molecular volume of water is taken to be30 AÊ 3. The molecular weights of the segments are58 for propylene oxide and 44 for ethylene oxideand these are used to calculate the number of PEOand PPO segments in the block copolymers.

SOLUBILIZATION Ð PREDICTIONS OFMOLECULAR THEORY WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

How Free Energy Contributions In¯uenceCapacity for Guest Molecules?

The various free energy contributions calculatedfor an illustrative PEO-PPO diblock copolymerwith benzene as the guest molecule are plottedagainst the aggregation number g in Fig. 3. Theremaining two independent variables D and � arenot kept constant but are chosen to be those valueswhich minimize the free energy of solubilizationper molecule ��og for each given value of g. Theimportance of each free energy contribution to


32 / Nagarajan

equilibrium solubilization behavior can be under-stood from Fig. 3.

The formation of micelles in preference to thesingly dispersed state of the copolymer occursbecause of the large negative free energy contribu-tion arising from a change in the state of dilution ofthe solvent incompatible A block following solubi-lization (curve a). In the absence of the solubilizate,this free energy contribution is a constant indepen-dent of the size of the micelle and hence does notgovern the aggregation number of micelle. How-ever, in the presence of the solubilizate, this freeenergy also accounts for the swelling of the Ablocks by the solubilizate J. This free energydecreases with an increase in the aggregationnumber g and is thus favorable to the growth ofthe aggregates. A second contribution favorable tothe growth of the aggregates is provided by theinterfacial energy (curve b). In general, the surfacearea per molecule of the micelle decreases with anincrease in the aggregation number. Consequently,the positive interfacial free energy between themicellar core and the solvent decreases withincreasing aggregation number of the micelle andthus this contribution promotes the growth of themicelle. The changes in the state of deformation ofthe A and the B blocks (curves c and d) and thechange in the state of dilution of the B block (curvee) provide positive free energy contributions thatincrease with increasing aggregation number of themicelle. Therefore, these factors are responsible for

limiting the growth of the micelle. More interest-ingly, when solvent W is a very good solvent for theB block, the positive free energy contributionsresulting from changes in the states of dilutionand of deformation of the B block outweigh thecontribution resulting from the changes in the stateof deformation of the A block. Under such con-ditions, the B block related free energy contribu-tions strongly influence the structural propertiesof the equilibrium micelles. The free energy oflocalization (curve f) is practically independent of gand thus has little influence over the determinationof the equilibrium aggregation number. The netfree energy of the micelle per molecule is shown bycurve g. As one would expect, this free energy isnegative and shows a minimum at the equilibriumaggregation number.

In Fig. 4, the various free energy contributionsare plotted as a function of the volume fraction ofthe solubilizate � in the micellar core. The remain-ing independent variables g and D/R are keptconstant at the values corresponding to the globalminimum of the free energy (i.e. with respect to g,Dand �) per molecule of the micelle. While Fig. 3helps illustrate how the equilibrium aggregationnumber is influenced by various free energycontributions, Fig. 4 reveals the importance ofvarious contributions in determining the equi-librium uptake of the solubilizate.

Curve a shows the large negative free energy

FIGURE 3. Dependence of various free energycontributions on the aggregation number g for a PEO-PPOdiblock copolymer. The other independent variables D and� assume values that minimize the free energy ofsolubilization of benzene for each value of g.

FIGURE 4. Free energy contributions as a function of thevolume fraction � of benzene in the micelle core for a PEO-PPO diblock copolymer. The other independent variables Dand g are assigned values corresponding to the globaloptimum of the free energy of solubilization of benzene forthis block copolymer molecule.



contribution provided by the change in state ofdilution of block A. This free energy decreases withan increase in the volume fraction of the solubili-zate. This factor thus favors the uptake of thesolubilizate within the micelle. One may note thatthe magnitude of this contribution is larger if thesolubilizate±core block interaction parameter wAJ issmaller and if the molecular size of the solubilizateis smaller. Correspondingly, the capacity of themicelles for such solubilizates (i.e. the equilibriumvalue of �) will be larger. When micelles incorpo-rate solubilizates, the radius of the micellar coreand the interfacial area per molecule of the micellarcore both increase. The former increases thepositive free energy contribution arising from theincreased deformation of the A block (curve c). Thelatter increases the positive free energy contribu-tion associated with the micellar core±solventinterfacial free energy (curve b). Thus both thesefactors serve to restrict the swelling of the micellarcore by the solubilizate and consequently, theextent of solubilization. One may note that theincrease in the positive interfacial free energyaccompanying the uptake of solubilizates by themicelles (shown by curve b) is also dependent onthe solubilizate±solvent interfacial tension �JW.Therefore, given two solubilizates, the micellarcapacity � will be larger for the solubilizateassociated with a lower solubilizate±solvent inter-facial tension �JW. Further, the increase in theamount of solubilizate within a micelle of specifiedaggregation number also changes the state ofdeformation (curve d) as well as the state ofdilution (curve e) of the B block in the shell regionof the micelle. Of these two positive free energycontributions, the former increases with �, thusdisfavoring solubilization while the latter decreaseswith increasing �, thus favoring increased solu-bilization. The free energy of localization (curve f)is practically independent of � and thus has littleinfluence over the nature and extent of solubiliza-tion. The net free energy of the micelle per moleculeis represented by curve g. The minimum in this netfree energy occurs at the equilibrium value for �.All the results described in the following sectionscan be interpreted in terms of the above-describedfree energy variations accompanying solubiliza-tion.

Size and Composition Dispersion of Aggregates

The calculated size and composition distribution(namely, the dispersion in the values of variables gand j around their most probable values) is shownin Fig. 5 for benzene solubilized within diblockPEO-PPO copolymer micelles. In constructing Fig.5, the volume fraction of the solubilizate within themicellar core � is chosen as the independentvariable in place of j. The point enclosed by theclosed curves corresponds to the most populousmicelles, that is those aggregates whose concentra-tion in the solution is the largest. The three closedcurves surrounding this point are the loci ofmicellar sizes and compositions corresponding to

which the micellar concentrations are respectively,one, two and three orders of magnitude smallercompared to the concentration of the most popu-lous micelles. One can observe that the aggregateconcentrations fall off very rapidly when the valuesof g and � deviate even slightly from those of themost populous micelles represented by the centralpoint in Fig. 5.

Similar numerical calculations of the aggregatesize and composition distributions for manysystems considered here indicate that the micellesare virtually monodispersed both in relation to thenumber of constituent block copolymer moleculesas well as the number of solubilizate molecules.Therefore, it is quite satisfactory to simplify thecalculations by invoking the pseudophase approxi-mation and then estimating the micellar character-istics by the minimization of the free energy permolecule ��og of an isolated micelle with respect tothe three independent variables R, D and �.

Predictions for PEO-PPO Diblock CopolymerMicelles

The predicted micellar structural parameters andthe solute uptake capacity in diblock PEO-PPOcopolymer E200P64 are summarized in Table 4. Alsoshown within the parenthesis in Table 4 are themeasured solubilization capacities from Table 1expressed as volume fractions within the micellecore. In general, the agreement between the

FIGURE 5. Distribution of the aggregation number g andthe solubilization capacity � in the equilibrium micelles. Theclosed curves are loci of sizes and compositions ofaggregates present at constant concentrations. Thecalculations are for benzene solubilized in a PEO-PPOdiblock copolymer.


34 / Nagarajan

experimental and measured values of � are reason-ably satisfactory for all the solubilizates. Whereasthe theory permits the prediction of all thestructural features of the micelles such as g, R, Dand the CMC, experimental data for these variablesare currently not available and hence the corre-sponding comparisons have not been possible.

Solubilization is found to increase the micellarcore radius and decrease the CMC. The larger thesolubilization capacity, the more significant are thechanges in R and the CMC. The increase in the coreradius R results not only from the incorporation ofthe solubilizate but also because of the increasingnumber of block copolymer molecules that areaccommodated within a micelle. This increase in gis more dramatic for solubilizates whose uptake bythe micelles is large. The shell thickness D is notvery much affected by solubilization.

The various contributions to the free energy ofsolubilization corresponding to the equilibriumaggregate are listed in Table 5 for the differentsolubilizate molecules. The extent to which theguest molecules are solubilized within the aggre-gates is reflected by the various free energycontributions. First, the negative free energy con-tribution due to the core block±solubilizate mixing

is largest for the guest molecule that is solubilizedmost. The magnitude of this contribution is larger ifthe solubilizate±core block interaction parameterwAJ is smaller and if the molecular size of thesolubilizate is smaller. Second, the larger theuptake of guest molecules, the larger the increasein the radius of the micelle core and the core±corona interfacial area per molecule. Consequently,the positive free energy contribution due to thedeformation of the A block and the free energy offormation of the core±corona interface both in-crease. Third, the larger the uptake of guestmolecules, the larger is the volume of the coronaregion of the aggregate which also becomes moredilute in polymer segments. Correspondingly, thefree energy contribution due to the deformation ofthe B and the dilution of the corona region increase.Lastly, the free energy of localization is practicallyindependent of the uptake of the guest molecules.

Predictions for PEO-PPO-PEO TriblockCopolymers

Table 6 lists the calculated results for the PEO-PPO-PEO triblock copolymer, E100P64E100, in water at25°C. A comparison with the results for diblock

TABLE 4. Predicted Core Radius, Corona Thickness, Aggregation Number, Solubilization Capacity and CMC of E200P64Diblock Copolymer Micelles at 25°C

Solubilizate R (AÊ ) D (AÊ ) g �a ÿln XCMC

Benzene 92.7 135 279 0.478 (0.51) 64.3Toluene 83.3 132 235 0.393 (0.40) 58.3o-Xylene 79.4 131 216 0.357 (0.33) 56.0Ethyl benzene 76.2 130 203 0.316 (0.41) 54.6Cyclohexane 69.8 128 180 0.211 (0.18) 52.5Hexane 58.1 120 122 0.069 (0.08) 48.3Heptane 57.2 120 119 0.057 (0.08) 47.8Octane 57.0 119 117 0.053 (0.08) 47.6Decane 55.7 118 112 0.036 (0.06) 47.2None 53.2 116 101 Ð 46.6

a Values in parenthesis are experimental data from ref. 4 summarized in Table 1.

TABLE 5. Calculated Contributions to the Standard Free Energy Change on Aggregation (in kT) for E200P64 DiblockCopolymer Micelles at 25°C

Solubilizate ��og�A;def ��o

g�A;dil ��og�B;def � ��o

g�B;dil ��og�int ��o

g�loc

Benzene 1.87 ÿ116.10 17.56 27.63 4.74Toluene 1.42 ÿ108.26 17.20 26.53 4.79o-Xylene 1.24 ÿ104.87 16.89 25.94 4.81Ethyl benzene 1.11 ÿ103.03 16.80 25.65 4.83Cyclohexane 0.86 ÿ100.12 16.72 25.17 4.89Hexane 0.45 ÿ91.19 14.82 22.70 4.94Heptane 0.42 ÿ90.39 14.67 22.51 4.94Octane 0.41 ÿ90.07 14.62 22.46 4.95Decane 0.37 ÿ89.04 14.38 22.17 4.95None 0.30 ÿ87.23 13.87 21.57 4.96



copolymers of the same molecular weight andcomposition (presented in Table 4) shows that themicellar core radius R, aggregation number g, andthe shell thickness D are all considerably smallerfor the case of triblock copolymers. The solubiliza-tion capacities are also somewhat reduced. TheCMCs are significantly larger for the triblockcopolymer compared to the diblock copolymer.For identical copolymer molecular weight andcomposition, the aggregation numbers of triblockcopolymer micelles are smaller than those of thediblock copolymer micelles. This is because the Ablock of the diblock copolymer is treated as beingequivalent to two A chains of half the molecularweight each in the triblock copolymer. So is the casewith the B block in the micellar corona. Effectively,the behavior of triblock copolymer E100P64E100 is

equivalent to that of the diblock copolymer E100P32

having half its molecular weight [30].

How the Block Size and Composition ofCopolymer In¯uences Uptake of GuestMolecules?

To investigate how the size and composition of theblock copolymer affect the uptake of guest mol-ecules, we have examined [39] the aggregation andsolubilization properties of the family of Pluronictriblock copolymers, EXPYEX. The calculated aggre-gation properties of the block copolymers in theabsenceof solubilizate are summarized inTable 7. Inperforming these calculations, � appearing in vari-ous free energy expressions is set equal to 0, thesolubilizate-related free energy terms are not rele-

TABLE 6. Predicted Core Radius, Corona Thickness, Aggregation Number, Solubilization Capacity and CMC ofE100P64E100 Triblock Copolymer Micelles at 25°C

Solubilizate R (AÊ ) D (AÊ ) g �a ÿln XCMC

Benzene 55.2 77.1 72 0.366 (0.51) 37.2Toluene 50.2 75.7 61 0.279 (0.40) 32.0o-Xylene 48.3 75.0 57 0.245 (0.33) 30.1Ethyl benzene 46.7 74.5 54 0.208 (0.41) 29.0Cyclohexane 43.7 73.7 49 0.124 (0.18) 27.2Hexane 39.3 71.0 39 0.039 (0.08) 24.5Heptane 38.9 70.8 38 0.031 (0.08) 24.2Octane 38.7 70.7 38 0.027 (0.08) 24.1Decane 38.2 70.3 37 0.016 (0.06) 23.8None 37.4 69.8 35 Ð 23.5

a Values in parenthesis are experimental data from ref. 4 summarized in Table 1.

TABLE 7. Predicted Core Size R (AÊ ), Corona Thickness D (AÊ ), Shape and/or Aggregation Number g of Solubilizate-FreeMicelles

Trade name Structure R (AÊ ) D (AÊ ) g for Spheres Shape

L62 E6P35E6 7.5 19.2 LL63 E9P32E9 14.5 13.9 LL64 E13P30E13 34.1 (38 to 46) 16.6 (37 to 44) 57 (39 to 70) SP65 E19P29E19 30.7 22.1 43 SF68 E77P29E77 21.3 (25) 51.0 (53) 15 (22) SP84 E19P43E19 42.2 22.5 75 SP85 E26P40E26 36.3 (37) 28.3 (36) 53 (57, 37 to 78) SF88 E104P39E104 25.0 63.9 17 SF98 E118P45E118 26.7 70.3 18 S

P103 E17P60E17 38.6 21.5 CP104 E27P61E27 51.0 29.8 94 SP105 E37P56E37 43.9 37.3 65 SF108 E133P50E133 28.3 (25.0) 76.6 (150) 20 (13) SP123 E20P70E20 42.1 24.4 CF127 E100P64E100 37.5 70.2 35 (15 to 45, 30) S

Sources of experimental data are cited in ref. 39 where the experimental conditions and methods are alsomentioned.


36 / Nagarajan

vant and the free energy minimization is done withrespect to the two independent variables R and D.The aggregate shape that yields the smallest freeenergy of aggregation is taken to be the equilibriumshape. The equilibrium aggregate morphology isdenoted in Table 7 by the symbols L, C, and S,whichrefer to lamellar, cylindrical and spherical aggre-gates. Also given are the dimensions of the core (R)andthecorona(D),andtheaggregationnumber(g) inthe case of spherical aggregates. The numericalvalues within brackets provided in Table 7 are someavailable experimental data.

The calculations show that lamellar aggregatesare favored when the ratio of PEO to PPO is smallwhereas spherical aggregates are favored when thePEO to PPO ratio is large. Typically, for blockcopolymers containing 40 wt% or more PEO, onlyspherical aggregates form at 25°C. For blockcopolymers containing 30 wt% PEO, cylindricalaggregates are possible. Block copolymers contain-ing 20 wt% or less PEO generate lamellae. One maynote that if the temperature is increased, the PEO±water interaction parameter wBW would increase. Inthe framework of the free energy model, thisincrease would lead to a shifting of the shapetransitions to higher PEO weight percent. Forexample, the P85 block copolymer which formsspherical aggregates at 25°C will generate cylind-rical aggregates at higher temperatures (corre-

sponding to a larger value for wBW), as has beenobserved experimentally [40, 41].

The predicted volume fractions of hydrocarbonsolubilized in aggregate core are summarized inTable 8 for six hydrocarbons and 15 Pluronic blockcopolymers. The aggregate shape is indicated asbefore by the symbols L, C and S. Also shownwithin parenthesis are experimentally measuredsolubilization capacities available in the literature(also expressed as volume fractions in the micellecore). In general, the agreement between theexperimental and measured values of � is reason-ably satisfactory. We do observe that for F127, theexperimental values, determined in our earlierstudies [2], are consistently larger than the pre-dicted values. We discuss later, various factors thatmay be responsible for such a disagreement. Ingeneral, the calculated results show that wheneverlamellar aggregates are formed, the solubilizationcapacity of the aggregates is the largest. Thesolubilization capacity progressively diminishesas we move from lamellar to cylindrical and thento spherical aggregates.

How Uptake of Guest Molecules Causes ShapeTransitions of Polymeric Aggregates?

Detailed structural information, namely the coredimension R and the corona thickness D, predicted

TABLE 8. Volume Fraction of Solubilized Hydrocarbon in Micelle Core and Aggregate Morphology

Trade name Benzene Toluene Xylene Cyclohexane Hexane Decane

L62 0.384 (L) 0.315 (L)(0.34)

0.287 (L) 0.179 (L) 0.064 (L) 0.033 (L)

L63 0.398 (L) 0.327 (L) 0.297 (L) 0.172 (L) 0.059 (L) 0.028 (L)L64 0.414 (L) 0.236 (C)

(0.403)0.207 (C)

(0.39)0.107 (C) 0.029 (C) 0.011 (S)

P65 0.268 (S) 0.195 (S)(0.46)

0.167 (S) 0.08 (S) 0.025 (S) 0.009 (S)

F68 0.203 (S) 0.13 (S) 0.105 (S) 0.038 (S) 0.011 (S) 0.003 (S)P84 0.481 (L) 0.301 (C)

(0.40)0.271 (C) 0.153 (C) 0.045 (S) 0.02 (S)

P85 0.32 (S) 0.243 (S) 0.213 (S) 0.11 (S) 0.036 (S) 0.014 (S)F88 0.248 (S) 0.167 (S) 0.139 (S) 0.056 (S) 0.016 (S) 0.004 (S)F98 0.268 (S) 0.185 (S) 0.155 (S) 0.065 (S) 0.019 (S) 0.006 (S)

P103 0.507 (L)(0.441)

0.442 (L)(0.375)

0.415 (L)(0.305)

0.278 (L) 0.074 (C) 0.04 (C)

P104 0.538 (L) 0.362 (C)(0.374)

0.331 (C) 0.177 (S) 0.061 (S) 0.031 (S)

P105 0.379 (S) 0.301 (S) 0.269 (S) 0.151 (S) 0.05 (S) 0.023 (S)F108 0.286 (S) 0.201 (S) 0.169 (S) 0.074 (S) 0.022 (S) 0.007 (S)P123 0.531 (L) 0.468 (L)

(0.382)0.442 (L) 0.305 (L) 0.083 (C) 0.048 (C)

F127 0.366 (S)(0.51)

0.279 (S)(0.40)

0.245 (S)(0.33)

0.124 (S)(0.18)

0.039 (S)(0.08)

0.016 (S)(0.06)

Sources of experimental data are cited in ref.39 where the experimental conditions and methods are alsomentioned.



TA

BLE

9.

Cor

eSi

zeR

(AÊ),

Cor

ona

Thic

knes

sD

(AÊ),

Shap

ean

dA

ggre

gatio

nN

umbe

rg

(inca

seof

Sphe

res)

ofM

icel

les

Con

tain

ing

Solu

biliz

edH

ydro

carb

ons

Trad

ena

me

Benz

ene

Tolu

ene

Xyle

neC

yclo

hexa

neH

exan

eD

ecan

e

L62

24

.6,

8.7

(L)

23

.3,8

.88

(L)

22

.6,

8.9

2(L

)2

1.3

,9

.23

(L)

18

.8,

9.2

2(L

)1

8.1

,9

.21

(L)

L63

21

.8,

13

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)2

0.3

,1

3.7

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19

.7,

13

.7(L

)1

7.9

,1

4.1

(L)

15

.6,

14

.0(L

)1

5.0

,1

4.0

(L)

L64

19

.3,

18

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)2

9.2

,1

7.6

(C)

28

.4,

17

.6(C

)2

6.4

,1

7.8

(C)

35

.2,

16

.7(S

61

)3

4.5

,1

6.6

(S5

9)

P65

39

.1,

22

.5(S

65

)3

6.7

,2

2.6

(S5

9)

35

.8,

22

.5(S

56

)3

3.6

,2

2.6

(S5

2)

31

.6,

22

.3(S

46

)3

1.0

,2

2.2

(S4

4)

F68

26

.0,

54

.4(S

21

)2

4.3

,5

3.3

(S1

9)

23

.7,

52

.9(S

18

)2

2.4

,5

2.0

(S1

6)

21

.6,

51

.3(S

15

)2

1.4

,5

1.1

(S1

5)

P84

26

.1,

25

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)3

8.7

,2

3.9

(C)

37

.5,

23

.9(C

)3

4.5

,2

4.4

(C)

44

.3,

22

.7(S

83

)4

3.1

,2

2.6

(S7

9)

P85

49

.1,

29

.0(S

88

)4

5.8

,2

9.0

(S8

0)

44

.4,

29

.0(S

75

)4

1.3

,2

9.1

(S6

8)

37

.9,

28

.5(S

57

)3

7.0

,2

8.4

(S5

4)

F88

32

.0,

69

.2(S

27

)2

9.6

,6

7.7

(S2

4)

28

.7,

67

.0(S

22

)2

6.8

,6

5.7

(S2

0)

25

.5,

64

.4(S

18

)2

5.1

,6

4.1

(S1

7)

F98

35

.1,

76

.8(S

31

)3

2.3

,7

5.0

(S2

6)

31

.2,

74

.2(S

25

)2

9.0

,7

2.7

(S2

2)

27

.4,

71

.1(S

19

)2

6.9

,7

0.6

(S1

9)

P10

33

6.2

,2

2.3

(L)

33

.7,2

2.7

(L)

32

.6,

22

.8(L

)2

9.3

,2

3.8

(L)

42

.0,

21

.7(C

)4

0.5

,2

1.7

(C)

P10

43

4.5

,3

3.6

(L)

50

.4,3

1.7

(C)

48

.6,

31

.7(C

)6

2.3

,3

0.7

(S1

42

)5

4.7

,3

0.2

(S1

09

)5

2.9

,3

0.0

(S1

02

)P1

05

63

.8,

38

.6(S

12

5)

59

.1,3

8.7

(S1

12

)5

7.2

,3

8.6

(S1

06

)5

2.5

,3

8.8

(S9

5)

46

.6,

37

.8(S

74

)4

5.1

,3

7.5

(S6

9)

F10

83

8.1

,8

4.2

(S3

4)

34

.9,8

2.2

(S2

9)

33

.7,

81

.3(S

27

)3

1.1

,7

9.5

(S2

4)

29

.1,

77

.4(S

21

)2

8.6

,7

6.8

(S2

0)

P12

34

0.9

,2

5.1

(L)

38

.1,2

5.7

(L)

36

.8,

25

.8(L

)3

3.1

,2

7.0

(L)

46

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24

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)4

4.5

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4.6

(C)

F12

75

5.2

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7.1

(S7

2)

50

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5.7

(S6

1)

48

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75

.0(S

57

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3.7

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39

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71

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39

)3

8.2

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0.3

(S3

7)


38 / Nagarajan

for the various equilibrium aggregates containingsolubilizates are summarized in Table 9. Solubiliza-tion is found to increase the micellar core radiusand also the aggregation number g; the larger thesolubilization capacity, the more significant are thechanges in R and g. Results for P65, P85, P105 andF127 illustrate this point, where spherical aggre-gates are present for all the solubilizates investi-gated, and therefore the influence of solubilizationon aggregate structure can be compared in theabsence of a shape change. In contrast, the coronathicknessD is not very much affected by solubiliza-tion. The core dimension is also very stronglyinfluenced by the equilibrium shape of the aggre-gate. In general, the core dimension R decreasesappreciably, whereas the corona thickness D onlymarginally increases, when the aggregate morphol-ogy changes from sphere to cylinder to lamellae.

To understand the origin of the solubilizate-induced transition in aggregate shapes, one canlook at the various free energy contributions. InTable 10, for illustrative purposes, the free energycontributions to micellization (the case of nosolubilizate) and solubilization in the spherical,cylindrical and lamellar aggregates are comparedfor P84 block copolymer. Also shown are theaggregate structural parameters R, D and � thatyield the lowest free energies for a given aggregateshape. The first entry of free energies in eachsubsection of Table 10 gives the results predictedfor the spherical aggregates. The second entry offree energies in each subsection of Table 10provides the difference between the free energy

contributions between a cylinder and a sphere, anegative value representing a contribution favoringthe cylinder over the sphere. Similarly, the thirdentry of free energies provides the difference in thefree energy contributions between a lamalla and asphere with a negative entry indicating a pre-ference for lamellar aggregates. One can considerthe various contributions with respect to aggregateshape by first reviewing the results for aggregationin the absence of any solubilizate. We note that thefree energy contributions associated with A blockdeformation (A,def) and formation of interface (int)provide negative energy differences indicating thatthese terms favor non-spherical aggregates. Thedecrease in the A block deformation contributionarises because of the smaller core dimensionconsistent with cylindrical and lamellar geome-tries. The decrease in the interfacial free energyarises from the smaller area per molecule consistentwith the non-spherical geometries. These negativefree energy differences are, however, overcompen-sated by the positive free energy differencesassociate with the corona block deformation(B,def) and mixing (B,dil) terms. The corona blockdimensions are not significantly altered by achange in the aggregate shape. Therefore, theconcentration of the polymer segments in thecorona region increases when the sphere changesto a cylinder or a lamella leading to a larger freeenergy of mixing contribution for the corona.Similarly, the stretching of the corona blockkeeping the concentration in the corona regionuniform costs a larger free energy for the lamellar

TABLE 10. Difference in Standard Free Energies per Molecule (in units of kT) Between Non-Spherical and SphericalAggregates for the Solubilization of Hydrocarbons in P84 Block Copolymer

Shape

Contributions to ��og� Total

��og� � R (AÊ ) D (AÊ )A,def A,dil int B,def B,dil

SolubilizateÐ benzeneS 4.90 ÿ83.83 22.81 4.64 1.15 ÿ39.57 0.338 57.0 22.5C-S ÿ0.61 ÿ1.02 ÿ0.41 0.40 0.99 ÿ0.66 0.376 41.5 23.6L-S ÿ1.58 ÿ3.57 ÿ0.7 1.24 3.59 ÿ1.03 0.481 26.1 25.6SolubilizateÐ tolueneS 4.28 ÿ79.16 21.8 4.75 1.24 ÿ36.32 0.265 53.6 22.7C-S ÿ0.59 ÿ0.9 ÿ0.34 0.43 1.05 ÿ0.41 0.301 38.7 23.9L-S ÿ1.54 ÿ3.17 ÿ0.41 1.34 3.86 ÿ0.09 0.41 24.0 26.0SolubilizateÐ cyclohexaneS 3.49 ÿ74.48 20.59 4.94 1.39 ÿ33.28 0.132 48.7 23.0C-S ÿ0.65 ÿ0.62 ÿ0.32 0.49 1.14 ÿ0.03 0.153 34.5 24.4L-S ÿ1.74 ÿ2.38 ÿ0.25 1.62 4.47 1.52 0.221 19.8 26.9SolubilizateÐ hexaneS 2.82 ÿ69.99 19.03 4.83 1.20 ÿ31.36 0.045 44.3 22.7C-S ÿ0.66 ÿ0.19 ÿ0.40 0.46 1.10 0.22 0.051 30.6 24.0L-S ÿ1.79 ÿ0.68 ÿ0.55 1.51 4.41 2.54 0.072 16.1 26.5No solubilizateS 2.52 ÿ67.95 18.28 4.76 1.09 ÿ30.57 0 42.2 22.5C-S ÿ0.65 0 ÿ0.42 0.43 1.07 0.33 0 28.8 23.8L-S ÿ1.74 0 ÿ0.62 1.42 4.32 2.95 0 14.5 26.2



and cylindrical aggregates compared to spheres.Since the overall differences in the free energy ispositive for cylinders and lamella, spherical aggre-gates are favored in the solubilizate-free system.

When the solubilizate is added, the abovegeneral features remain unaltered but there is anadditional negative free energy difference due tothe mixing of the solubilizate with the core block(A,dil). The lower free energy of A block±solubili-zate mixing in lamellae with respect to cylinder,and for cylinder compared to sphere, arises fromthe increasing volume fraction of the solubilizate asthe aggregate shape changes from sphere tocylinder to lamella. This negative free energydifference is small when hexane is the solubiliate(consistent with a larger value for wAJ). Therefore,the spherical aggregates continue to be the equi-librium structures. When the solubilizate is cyclo-hexane, the difference in the free energy termbecomes negative for cylinders indicating thefavorable formation of cylindrical aggregates.When the solubilizate is toluene, the difference inthe free energy term is negative both for cylindricaland lamellar aggregates. However, the magnitudeof the term for cylinder is larger (implying a lowertotal free energy), thus cylindrical aggregates arefavored.When benzene is solubilized, as before, thedifference in the free energy term is negative bothfor cylindrical and lamellar aggregates. However,the magnitude of the term for a lamella is largerthan that for the cylinder and hence, lamellaraggregates are favored.

One can observe that the main contributionchanging the behavior of the free energies corre-sponding to the different solubilizates is the Ablock±solubilizate mixing free energy. As theinteraction parameter wAJ decreases from hexaneto cyclohexane to benzene, the mixing free energyprogressively leads to higher volume fraction of thesolubilizate and a structural change from sphere tocylinder to lamella.

A comprehensive summary of the equilibriumaggregate shapes for various EXPYEX block copo-lymers are summarized in Fig. 6, both in theabsence of any solubilizate and in the presence offour hydrocarbons. In the Pluronic grids shown,the weight fraction of PEO increases as one movesto the right and the molecular weight of the PPOblock increases as one goes down. Therefore, themolecules in the bottom left section of the grid aremore hydrophobic while those on the top rightsection are more hydrophilic. In the absence of anysolubilizate, Pluronics in the 20 series (L62, L72,L92, L122) form lamellar aggregates, Pluronics inthe 30 series form either lamellae (L63) or cylinders(P103, P123) while Pluronics containing 40 wt% ormore of PEO form spherical aggregates. Thesolubilization of hexane does not lead to anyaggregate shape transitions. However, when thesolubilizate is cyclohexane, a cylinder to lamellatransition is predicted for P103 and P123 while asphere to cylinder transition is suggested for L64and P84. When toluene is the solubilizate, thesphere to cylinder transition is extended to P104.

FIGURE 6. Equilibrium aggregate morphologiesgenerated by various Pluronic block copolymers in theabsence of any solubilizate and in the presence of hexane,cyclohexane, toluene and benzene as the solubilizates. Theequilibrium structures are those that are predicted to occurin the limit of saturation solubilization of the hydrocarbons.The horizontal lines denote lamellar aggregates, thehoneycomb represents cylindrical aggregates and thevertical lines refer to spherical aggregates.


40 / Nagarajan

Finally, for benzene as the solubilizate, the cylind-rical aggregates of P103 and P123 as well as thespherical aggregates of L64, P84 and P104 allundergo a predicted transition to lamellar aggre-gates. It is evident that by choosing an appropriatesolubilizate and the amount of solubilization, onecan induce a desired type of structural transition inthe Pluronic block copolymer system.

Approximations in Modeling and Consequences

The quantitative prediction of aggregate character-istics reviewed in this paper are influenced by thesimplifying assumptions that have been made inconstructing the model. Firstly, the model assumesa sharp core±corona interface and does not allowthe penetration of water or of the hydrophilic blockinto the hydrophobic core. It is of interest to relaxthis assumption and examine how the modelpredictions will be modified for the case of adiffuse interface. Secondly, for solubilizates such asbenzene which are also good solvents for PEO,there is the possibility that in addition to thesolubilizate being present in the micellar core, itcould also be present in the micellar corona. Thepresence of any solubilizate in the micellar coronacan alter the predicted micellar dimensions and thesolubilization capacity. The model presented heredoes not describe such a situation. Third, onlyapproximate estimates for the interfacial tension�agg characteristic of the core±corona interface areemployed in the present calculations. To obtainimproved estimates of �agg, future developments inthe treatment of a constrained interface betweentwo solutions are necessary. Fourth, the PEO±waterinteractions in corona region are described usingthe Flory model and taking a constant value for thePEO±water interaction parameter wBW. However, itis known [42] that accurate representation of theliquid±liquid phase equilibrium behavior of PEO±water systems requires a wBW that is dependent ontemperature, composition of the solution and themolecular weight of the polymer, if the Floryequation is used. Therefore, a more fundamentaltreatment of PEO±water system would be neces-sary to more fully account for the interactions in thecorona region. All of the above features would leadto changes in the magnitude as well as sizedependence of the free energy and thus have animpact on all the predicted structural features ofaggregates including the extent of solubilization.Finally, the PPO±water interactions are also repre-sented in the model using a constant value for wAW,obtained from vapor±liquid equilibrium studies.Small changes in the estimate for this interactionparameter will significantly affect the magnitude ofthe free energy, but not its dependence onaggregate size. Consequently, the calculated CMCcan be considerably modified by changes in wAW,but all the predicted structural features of aggre-gates would remain invariant. Considering theexperimental studies, all the results presented inthis work have been obtained on commercialsamples of Pluronics which may contain impurities

such as homopolymers PEO, PPO, and blockcopolymers of differing molecular weights andcompositions. In contrast, all the predictive calcula-tions have been performed for a hypothetical pureblock copolymer having the molecular weight andcomposition specified for the commercial product.

Having enumerated the limitations of thepredictive calculations as above, one can stillrecognize that the main advantage of the presenttheory is that it allows the prediction of all themicrostructural features of micelles containingsolubilizates with only minor computational effort.Another advantage of the theory is that all freeenergy contributions are given as explicit analyticalfunctions directly linked to physicochemicalchanges accompanying micellization and solubili-zation, and involving only molecular constants andgeometrical variables. Since our attempt has beento make predictions using well defined models andmolecular constants rather than to fit empiricallythe experimental data, we can take advantage of theobserved contradictions between the predictionsand the experiments to improve or modify the freeenergy expressions and to improve the estimatesfor the molecular constants. Also new physicaleffects such as those mentioned in the earlierparagraph which are not included in the modelpresently can be explored. However, to justifymeaningful future developments in the theory ofsolubilization, it is essential to have more detailedmeasurements of various microstructural charac-teristics of micelles than are presently available. Forexample, if we have the core and corona dimen-sions, aggregate shapes and the amount solubilizedfor at least a few block copolymers, we can test andimprove the model by stipulating that all of theavailable structural features should be predictedwell. It is hoped that such experimental studieswould be undertaken in the future and facilitate acritical evaluation and development of the theoryof solubilization.

Although the main theme of this paper has beenfocused on hydrophobic low-molecular-weightsolubilizates, the Pluronic block copolymer hasalso been used to solubilize both hydrophobic andhydrophilic enzymes [43] and also a variety ofpolar organic molecules like ketone, esters, alcoholsand aldehydes [3]. A theoretical treatment of thesolubilization behavior of such substances remainsto be developed. We had mentioned earlier thatboth PEO and PPO blocks may serve as locus ofsolubilization in the case of the hydrophobicaromatic solubilizates. Such a situation may bealso relevant to the solubilization of proteins andthe polar organic solubilizates. Further experimen-tal studies of solubilization of these types of solutesby Pluronics will be useful in the development oftheories of solubilization governing such solutes.

SOLUBILIZATION Ð SOMEAPPLICATIONS WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWThe ability of block copolymer aggregates to hostguest molecules can be exploited for many applica-



tions. Here we refer to a few recent studies wherethe solubilization has been taken advantage of forcarrying out chemical separations, metal nanopar-ticle synthesis, enzymatic biocatalysis and drugdelivery. The solubilization of naphthalene in threePEO-PPO-PEO triblock copolymers has been stu-died [19], with special emphasis on the influence oftemperature. It was found that the solubilization ofnaphthalene can be appreciably increased bymoderate increases in the temperature. The solubi-lization process was found reversible so that thesolubilizate which separates and crystallizes out ata lower temperature can be readily separated fromthe block copolymermicelles. This is very useful forthe recovery of solubilizates from the micelles andfor the recycling of the micelles in a process fortreating environmental pollutants like polycyclicaromatics.

Novel block copolymers have been synthesized[44] to allow the solubilization of metal salts inblock copolymer micelles having a polar core and anon-polar corona formed in toluene. The diblockcopolymers were synthesized starting from com-mercially available PS-polybutadiene block copoly-mers. The polybutadiene block was subjected toepoxidation and the resulting polyepoxide blockwas subsequently modified by different types ofring-opening chemical reactions to create blockswith metal-complexing side groups. The lipophilic-metalophilic diblock copolymer was used tosolubilize numerous metal salts of gold, silver,palladium, rhodium, copper and zinc. The possi-bility of preparingmetal nanocolloids through suchsolubilization route has been demonstrated [44]opening an important application of solubilizationby block copolymer micelles.

The solubilizates in block copolymer micellesneed not be limited to low molecular weighthydrophobic substances. Homopolymers havebeen solubilized in dilute solutions of blockcopolymer micelles [45]. Also, high-molecular-weight proteins have been solubilized in blockcopolymer systems [43]. The use of the microdo-main structure of block copolymer aggregates as aneffective enzymatic reaction medium was demon-strated by solubilizing enzymes within the hydro-philic domains of the block copolymer. Theimmobilized enzyme reacts with water-insolublesubstrates that are solubilized within the hydro-phobic domains acting as microreservoirs of sub-strates. It is possible to use the block copolymermicrodomains profitably also in case of water-soluble substrates, if the reaction product is water-insoluble. In this case, the product is continuouslyremoved from the hydrophilic domains by itstransfer into the hydrophobic domains acting asmicrosinks, thus the problem of product-inhibitionis avoided. Two enzymatic reactions were explored[43]. In the first example, the enzyme cholesteroloxidase was used to oxidize cholesterol to choles-tenone. The advantage derived from the increasedsolubility of cholesterol in the block copolymersystem (approximately 22,000 mM in the pure blockcopolymer compared to the aqueous phase solu-

bility of 4.7 mM) is obvious. The second enzymaticreaction involved horse radish peroxidase whichwas used to catalyze the oxidation of pyrogallol topurpurogallin by hydrogen peroxide. In this case,the substrate has considerable water solubilitywhile the reaction product has a lower watersolubility. The dissolution of purpurogallin in thehydrophobic microdomains removes it from wateras it forms, and helps overcome product-inhibitionfrom affecting catalytic activity. This reaction is ofinterest to the treatment of phenolic wastes and forthe synthesis of polymeric products. The stability ofthe enzyme cholesterol oxidase in the microdo-mains was even better than that in the aqueousmedium and the enzymatic activity in microdo-mains was retained over extended periods of time.

Block copolymer aggregates are finding impor-tant new applications in the area of drug delivery.Recent work in this area has been summarized inref. [46]. Applications such as tissue specific drugdelivery, delivery of anti-tumor agents, genetransfection, diagnostic imaging, delivery of drugsto central nervous system, use of biodegradableand thermo-responsive micelles for drug deliveryare among the topics that have been explored.

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