Large Internal Structures of Micelles of Triblock Copolymers with Small

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Large Internal Structures of Micelles of Triblock Copolymers with Small Insoluble Molecules in Their Cores Li Xing and Wayne L. Mattice* Institute of Polymer Science, University of Akron, Akron, Ohio 44325-3909 Received January 27, 1998. In Final Form: May 5, 1998

This Monte Carlo study on a simple cubic lattice investigates ternary mixtures of triblock copolymersolvent-solubilizate systems consisting of solvent molecules, singly dispersed copolymer molecules, singly dispersed solubilizate molecules, and micelles of all sizes containing various amounts of insoluble blocks and solubilizates. In analogy with low molecular weight surfactant systems, two model structures are considered, one for dissolution and the other for microemulsion. Various approaches are carried out to identify and differentiate these two models. Under weak solubilization conditions, solubilizates are uniformly dissolved in the micellar cores. A gradual transition to the second model, where solubilizates accumulate in the centers of micellar cores with surfaces decorated by copolymers, occurs as solubilization becomes strong, due to either the presence of large quantities of solubilizates or the high incompatibility between immiscible species. As a consequence the solubilization process takes place in two steps, with each step arising from a distinct mechanism. The Langmuir adsorption isotherm is a good approximation for weak solubilization, while significant deviation from this isotherm is seen at strong solubilization.

Introduction The solubility of predominantly hydrophobic molecules in aqueous solution is enhanced by the addition of surfactants. The added surfactant molecules self-assemble to form micelles or vesicles which, by providing a more compatible environment for the sparingly soluble molecules, increase their solubility. These small insoluble molecules are called solubilizates, and the related dissolution phenomena is called solubilization. This tendency for solubilization constitutes the basis on which surfactants can be exploited in controlled uptake and release of drugs, pollutants, and other compounds in many biological, pharmaceutical, and environmental systems. In a selective solvent which is good for one monomer but poor for the other monomer, block copolymers spontaneously form micelles with the poorly soluble block forming the core and the easily soluble block forming the corona. Block copolymers hold an unusual potential compared to conventional low molecular weight surfactants because of their ability to micellize in a variety of solvents and the option to custom design for a particular application by changing the molar mass of the blocks. In the past decade micellization and solubilization in block copolymer surfactant molecules have received increased attention from both a theoretical1-9 and experimental10-18 (1) Nagarajan, R.; Ganesh, K. J. Chem. Phys. 1989, 90, 5843. (2) Nagarajan, R.; Ganesh, K. Macromolecules 1989, 22, 4312. (3) Cogan, K. A.; Leermakers, F. A. M.; Gast, A. P. Langmuir 1992, 8, 429. (4) Shull, K. R.; Winey, K. I. Macromolecules 1992, 25, 2637. (5) Dan, N.; Tirrell, M. Macromolecules 1993, 26, 637. (6) Hurter, P. N., Scheutjens, J. M. H. M.; Hatton, T. A. Macromolecules 1993, 26, 5030. (7) Hurter, P. N., Scheutjens, J. M. H. M.; Hatton, T. A. Macromolecules 1993, 26, 5592. (8) Linse, P. Macromolecules 1994, 27, 2685. (9) Leerkamers, F. A. M.; Wijmans, C. M.; Fleer, G. J. Macromolecules 1995, 28, 3434. (10) Zhao, C.-L.; Winnik, M. A.; Riess, G.; Croucher, M. D. Langmuir 1990, 6, 514. (11) Wilhelm, M.; Zhao, C.-L.; Wang, Y.; Xu, R.; Winnik, M. A.; Mura, J.-L.; Riess, G.; Croucher, M. D. Macromolecules 1991, 24, 1033. (12) Cao, T.; Munk, P.; Ramireddy, C.; Tuzar, Z.; Webber, S. E. Macromolecules 1991, 24, 6300.

point of view. In addition to the knowledge about the extent to which a certain compound can be solubilized in a given surfactant solution, a focus of most experimental studies, the other aspect is to know the localization of solubilizate molecules within the micelles, which is of particular interest in understanding the nature of the catalytic activity of micellar systems. The microenvironment of the solubilizate molecules in surfactant micelles is extensively studied. Surfactant micelles can be pictured as having a highly nonpolar interior and a relatively polar interfacial region. The interior of the micelle is generally considered to be the locus of solubilization for very nonpolar solubilizates such as n-alkanes. Solubilizate molecules of relatively high polarity, such as alcohols, are believed to be solubilized in the interfacial region of the micelle so that their polar functional groups could retain their contact with water. For molecules which are only slightly polar, such as aromatic hydrocarbons, there is less agreement concerning their location in the micelle, although the issue has been addressed.19-22 In contrast, there have been very few studies of the localization of low mass substances in block copolymer micelles. From measuring the refractive index increments of the aqueous solution of poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) (PEO-PPO-PEO) (13) Cogan, K. A.; Capel, M.; Gast, A. P. Macromolecules 1991, 24, 6512. (14) Kiserow, D.; Prochazka, K.; Ramireddy, C.; Tuzar, Z.; Munk, P.; Webber, S. E. Macromolecules 1992, 25, 461. (15) Hruska, Z.; Piton, M.; Yekta, A.; Duhamel, J.; Winnik, M. A.; Riess, G.; Croucher, M. D. Macromolecules 1993, 26, 1825. (16) Alexandridis, P.; Holzwarth, J. F.; Hatton, T. A. Macromolecules 1994, 27, 2414. (17) Gadelle, F.; Koros, W. J.; Schechter, R. S. Macromolecule 1995, 28, 4883. (18) Procha´zka, K.; Martin, T. J.; Munk, P.; Webber, S. E. Macromolecules 1996, 29, 6518. (19) Hirose, C.; Sepulveda, L. J. Phys. Chem. 1981, 85, 3689. (20) Nagarajan, R.; Ruckenstein, E. Sep. Sci. Technol. 1981, 16, 1429. (21) Simon, S. A.; McDaniel, R. V.; McIntosh, T. J. J. Phys. Chem. 1982, 86, 1449. (22) Nagarajan, R.; Chaiko, M. A.; Ruckenstein, E. J. Phys. Chem. 1984, 88, 2916.

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with a small amount of o-xylene as the solubilizate in the PPO core, the refractive index increment decreased as the ratio of xylene to copolymer increased, despite the relation of the refractive indexes nwater < ncopolymer < nxylene.23 Thus it was deduced that o-xylene cannot be distributed homogeneously in the copolymer. This phenomenon was rationalized by assuming that a xylene core is surrounded by a copolymer layer, but no further investigations were carried out. Polar solubilizates, such as bromoacetonaphthone, may prefer sites on the surface of the micelle.15 If the solubilizate has some affinity for isolated chains that comprise the soluble corona, it can be expected that some of them will be bound to the micelle, but in an environment that is rich in solvent.12,14 In the present work, we will be concerned with nonpolar solubilizates that have no affinity for the soluble block. In our previous work24,25 we monitored triblock copolymer systems mixed with small solubilizates on a simple cubic lattice by the Monte Carlo method. Bulk properties critical to micellization and solubilization, such as aggregation number, micelle-solvent partition coefficient, and solubilization capacity, were extracted after equilibration, and their dependence on various system parameters was investigated.24 Geometric properties, specifically the size and shape of copolymer micelles with solubilized low-mass molecules, were obtained and compared with experiments.25 In this paper we focus on a microscopic picture regarding the locus of solubilizates in ABA triblock copolymer micelles when the solubilizate molecules are energetically equivalent to a segment of the insoluble block. This situation is of practical interest since monomer or homopolymer impurities are hard to eliminate from block copolymer synthesis. The transition between two characteristic model structures is identified, and the solubilization mechanism is deduced as the amount of solubilizates increases. Finally the applicability of Langmuir adsorption theory is examined under different conditions in light of the polymer concentration and interaction between incompatible species. Method The methodology employed in the simulation is essentially the same as that used in recent work.24,25 The mixture of triblock copolymer, solubilizate, and solvent is simulated on an L × L × L three-dimensional cubic lattice. L is chosen at 40 for all the simulations, and periodic boundary conditions are applied in the x, y, and z directions. Every chain of the symmetric triblock copolymer, ANABNBANA, has the same size, where each block contains NA beads of A or NB beads of B. Each insoluble small molecule, I, occupies one lattice site. No lattice site can be occupied by more than one bead. Vacant sites are considered to be solvent, S. For a given system containing NABA copolymer molecules and NI solubilizates, the solvent molecules occupy a total of NS ) L3 - NABA(2NA + NB) - NI sites. Only nearest neighbor interactions are considered. The reduced interaction energy is denoted by XY ) EXY/kBT, where X and Y represent any one in the pool of A, B, I, and S. In total there are 10 different XY pairs. For a reaction of

XX + YY f 2XY

(1)

the driving force determining the reaction direction is the energy difference XY - (XX + YY)/2, but not the individual energy terms. Taking the pairwise interactions of the same species as a reference, such a reaction is described by one independent interaction parameter, XY. Applying the same schema to our (23) Tontisakis, A.; Hilfiker, R.; Chu, B. J. Colloid Interface Sci. 1990, 135, 427. (24) Xing, L.; Mattice, W. L. Macromolecules 1997, 30, 1711. (25) Xing, L.; Mattice, W. L. Macromol. Theory Simul. 1997, 6, 553.

system of four different species, six independent energetics, specifically AB, AI, AS, BI, BS, and IS, are extracted. The compatibility between middle block and solvent, and between end block and solute, is achieved by the assignment BS ) AI ) 0. The interaction energies of the other pairs, namely, AB, AS, BI, and IS, are assigned positive values to favor separation of A blocks from B blocks and solvent and to give solubilizate molecules a preference for A blocks. Each simulation starts with some initial configuration where a certain number of copolymer and solubilizate molecules is randomly distributed on the cubic lattice; then the copolymer and solubilizate molecules are randomly selected to make a move subject to the restriction that double occupancy of any site is forbidden. Each small insoluble molecule moves at random to one of its six nearest neighbors, whereas each polymer molecule moves by reptation, kinkjump, or crankshaft, one adopted randomly at a time. Each attempted move is accepted or rejected according to the Metropolis rule. After a large number of moves (of order 5 × 108) the system eventually evolves into its equilibrium state. The physical quantities are then calculated by averaging over 100 replicas at an interval of 100 Monte Carlo steps. One Monte Carlo step attempts a move on every segment of the block copolymers and on every I and is thus equivalent to NABA(2NA + NB) + NI steps. Interchain aggregation is defined to occur whenever a bead of A on one chain is a nearest neighbor of a bead of A on another chain or an indirect neighbor of an A on another chain through I.24

Results and Discussion A representative snapshot of the triblock copolymers in a starting conformation is depicted in Figure 1a. As we turn on the energy, because of the selectivity of the solvent, the copolymer chains form micelles, with the two end A blocks composing the core and the middle B block making the shell. A representative equilibrium structure is shown in Figure 1b. The energetic penalties contributing to the aggregation in Figure 1b come from formation of nearest neighbors between a segment of A and a segment of B and between a segment of A and solvent S. Under this condition of strong segregation, very few A blocks disperse freely without participating in micellization. Most of the chains that are involved in the aggregates adopt the conformation of a loop, thereby placing both of their terminal blocks in the core. Some of the B blocks play the role of bridges, linking two micelles with their two end A blocks. Such dynamic bridges can lead to the formation of a transient physical network if the concentration is sufficiently high.26 Although the solubilizates are not shown because a two-dimensional projection of a threedimensional replica may be misleading, it is easy to imagine that they will be adsorbed in the tight cores since they are compatible only with A. Two Models for Micelle Structure. In analogy with the structural models for solubilization and microemulsions in systems involving low molecular weight surfactants, two kinds of aggregates for spherical shapes can be pictured, depending on how the solubilizates are located inside the block copolymer micelles. Figure 2a is typically used to represent solubilization while Figure 2b is used to denote droplet microemulsions. In Figure 2a the solubilizates and the A blocks constituting the micellar core are randomly and uniformly distributed throughout this core, whereas in Figure 2b a region of pure solubilizate I is surrounded by an inner shell region consisting of the solvent-incompatible A blocks and the solubilizates. The models differ in two ways: First, I’s are present as a pure fluid in the second model, but not in the first model. Second, (26) Nguyen-Misra, M.; Mattice, W. L. Macromolecules 1995, 28, 1444.

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Figure 2. Two models for the structure of micelles. (a) Uniform dissolution of I in A. (b) Stabilization of a microdroplet of I by putting A on the interface.

Figure 1. Snapshot of (a) a relaxed system where all ’s are zero and (b) a system in a stable state where the nonzero ’s are AB, AS (Dark: end (A) block. Light: middle (B) block. A15B10A15, VFABA ) 0.09, AB ) AS ) 0.45).

the insoluble block is found throughout the core in the first model, but not in the second model. It has been suggested that the thermodynamic equilibrium criterion favors the occurrence of the structure in Figure 2a, so that the second structure can be excluded.1 In another theoretical study it was assumed that homopolymer B is uniformly distributed throughout the cores in the system of A-B block copolymers in homopolymer A host to which B was added.27 It was predicted that once the threshold in free energy due to stretching and interfacial tension is reached, all the homopolymers are expelled from the cores into a second phase. Although homogeneously dispersed domains of PEO were revealed with polystyrene-poly(ethylene oxide) (PS-PEO) in PS hosts by tunneling electron microscope, they were interpreted as a separated macrophase rather than a microemulsion.27 (27) Whitmore, M. D.; Smith, T. W. Macromolecules 1994, 27, 4673.

All systems spontaneously tend to lower their free energy, which is a balance between enthalpy and entropy. Generally speaking the free energy for the structure in Figure 2b is higher than that in Figure 2a because of the entropic sacrifice that accompanies localizing A blocks at the interface. However, the entropy may be overweighed by the enthalpy if the latter makes the dominant contribution to the free energy. Both the enhancement of the unfavored interaction between solute and solvent and the addition of more insoluble molecules are feasible ways to strengthen the enthalpic contribution. It is conjectured that at the point when the enthalpy dominates the system, the second model may be favored. In the following sections we present results that assess the validity of this conjecture. Addition of Solubilizates. To resolve the controversy over spheric micelles, the two models have to be differentiated from one another in a well-defined manner, which can be done either quantitatively or qualitatively. In the quantitative way, the squared radius of gyration of A’s (Rg-A2) and I’s (Rg-I2) constituting the same micelles are calculated separately. The squared radius of gyration, defined as the average squared distance of every bead constituting the entity to the center of mass, measures the isotropic extension of the entity.25 While Rg-A2 and Rg-I2 are dealt with separately, both A’s and I’s are included in obtaining the center of mass. Therefore by comparing the magnitude of Rg-A2 and Rg-I2, information regarding the closeness or remoteness of two kinds of beads to the core of the micelles can be extracted. Values of Rg-A2 similar to Rg-I2 justify the validity of the first model, while Rg-A2 greater than Rg-I2 argues for the second model displayed in Figure 2b. The only variable in this section is the volume concentration of solubilizates, VFI, which varies from 0.0078 to 0.125. The rest of the parameters are fixed at NA ) 5, NB ) 10, AB ) AS ) 0.45, IS ) BI ) 0.8. A polymer volume fraction VFABA of 0.04 is achieved with 128 chains. Figure 3 depicts Rg2 for A’s, I’s, and complete micellar cores comprising both A’s and I’s, as the concentration of I’s

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Figure 3. Squared radius of gyration for A (Rg-A2), I (Rg-I2), and both A and I (Rg2) in the same micelles, as a function of the volume fraction of solubilizates, VFI (128 chains of A5B10A5, AB ) AS ) 0.45, IS ) BI ) 0.8).

Figure 5. Volume fraction distributions of A, B, and I at VFI ) 0.007 81, 0.0625, 0.125 (128 chains of A5B10A5, AB ) AS ) 0.45, IS ) BI ) 0.8).

Figure 4. Rg-A2 - Rg-I2, as a function of the volume fraction of solubilizates, VFI (128 chains of A5B10A5, AB ) AS ) 0.45, IS ) BI ) 0.8).

increases. To obtain good statistics, the data points are averaged over all the micelles in every replica, and further over 100 replicas. At low solute concentrations, Rg-A2 almost equals Rg-I2, implying the adoption of the first model. But as VFI increases, Rg-A2 grows faster and starts to depart from Rg-I2. The fact that Rg-A2 is substantially larger than Rg-I2 at high concentration of the solubilizates justifies the dominance of the second model. To see the effect more easily, the difference between Rg-A2 and Rg-I2 is plotted in Figure 4. Starting at the value close to zero, Rg-A2 - Rg-I2 increases almost monotonically as more insoluble molecules are added, showing a gradual transition from the first model to the second one. The qualitative or more visual approach is to examine the radial distribution functions of A’s and I’s with respect to the distance, r, from the center of the micelles. In Figure 5, the volume fraction distributions along the radial direction, originating from the core toward the periphery of each micelle, are plotted. Again the curves are averaged over every micelle at a certain step and further over 100 replicas. The horizontal axis is the distance of the spherical shell from the center of mass of the aggregate, expressed in units of the lattice spacing. The three panels in Figure 5 present the volume fractions of the three species, specifically A, B, and I, in successive spherical shells. The sum of these three volume fractions is 1 only if solvent is excluded from the spherical shell. Results are depicted for three typical solubilizate concentrations, VFI ) 0.0078, 0.0625, and 0.125. At the lowest VFI, the

distribution of A’s is localized around the center of the micelles, and the cores are swollen by solvent, with the volume fraction of solvent being about 0.3 at the center of mass. A segments are pushed away from the central core at VFI ) 0.0625 and even further away at VFI ) 0.125. The peak of the distribution of A shifts from 0.5 through 4.3 to 6.9 lattice units accordingly. Accompanying the departing of A’s toward corona, the distribution curve broadens, indicating the smearing around of A. Excluded by A segments, the occurrence of B’s (middle panel of Figure 5) is displaced toward larger r, forming the shells of the micelles at larger radii. As depicted in the bottom panel of Figure 5, the additional solubilizates accumulate in the central core region by squeezing out both the solvent molecules and the A blocks, forming microdroplets. A few A blocks are positioned at the solute-solvent interface, acting as intermediate agents. This picture is consistent with the two-dimensional schematic shown in Figure 2b. The other issue of concern is whether the micelles change shape during the expansion. Are they still approximately spherical or do they become more ellipsoidal? The answer to this question was sought by calculating the asphericity and acylindricity, which measure the roundness and elongation of the shape.25 These two measures detect no specific shape transition associated with the swelling of the micelles. As more insoluble molecules are added, the micellar cores expand isotropically. If we number the consecutive beads of an A block from 1 to NA, having 1 as the bead at the free end and NA as the bead connected to B, it is of interest to learn the relative locus of A1 and ANA within the micellar core. For the highest concentration of I studied (VFI ) 0.125), the distributions of A1 and ANA are plotted in Figure 6, together with the distribution of all A segments constituting the aggregation. The free ends are displaced toward the I-rich inner core, while the beads at the A-B junction are more adjacent to core-shell junctions. The monomers at the junction of two distinct blocks are confined in a narrow interfacial area. If the size of the aggregating core becomes

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Figure 6. Volume fraction distributions of the first and the NAth bead of the A block, together with the total A in the core of micelles (128 chains of A5B10A5, AB ) AS ) 0.45, IS ) BI ) 0.8, VFI ) 0.125).

Figure 7. Distance from the A bead at the A-B junction to the center of the micellar core (rNA) and the end-to-end distance of the A blocks in micelles (Re) as a function of the volume fraction of solubilizates, VFI. Standard deviations are depicted as half error bars (128 chains of A5B10A5, AB ) AS ) 0.45, IS ) BI ) 0.8).

big enough to surpass the end-to-end distance of the sticky A blocks, the free ends can no longer reach the center core, hence leaving there a pool of pure I. This point is addressed in more detail as follows. Figure 7 shows that the average end-to-end distance of an A block in an aggregate is nearly constant (in the range 2.1-2.4) for the VFI considered here. In contrast, the distance from the A bead at the A-B junction to the center of mass of the aggregate increases as VFI increases. These two distance are comparable at small VFI, where the free ends of A are able to touch the center of mass of the core. The core expands as more I’s are added, producing a particle in which the free end of A can no longer reach the center. Hence the micellar cores are split into two layers, with the inner layer composed of pure I and the outer one a mixture of I’s and A’s. The interaction penalty, IS (0.8), is higher than AS (0.45), in the simulation described here, so it is not surprising that the system has a stronger avoidance of I-S contacts than A-S contacts and therefore places A at the interface as a mediating agent. Mechanism of Solubilization. Solubilization takes places in two steps in the simulations: (1) Solubilization initially takes place by displacing the solvent molecules from the micellar core. Unlike the conventional surfactant micelles, substantial quantities of solvents are present in the micellar core of the small ABA triblock copolymers, which was predicted

Xing and Mattice

Figure 8. Distance from the A bead at the A-B junction to the center of micellar core (rNA) as a function of the interaction parameter, IS ) BI. The horizontal line at 2.2 is the average end-to-end distance of the A blocks in the aggregates (A5B10A5, AB ) AS ) 0.45, I:A ) 0.96).

by Hurter et al.6,7 and Linse and Malmsten.28 At this stage the micellar structure complies with the model drawn in Figure 2a, which is the uniform dissolution of I in A. (2) Subsequent solubilization takes place by further displacing the solvent molecules and by excluding the insoluble blocks away from the center of the core. If the A blocks cannot stretch into the centers of the cores of the swollen micelles, solubilizates will accumulate in the centers. The corresponding micellar structure is as shown in Figure 2b. The distinct characteristic of this structure is that there is a region where only pure solubilizate molecules are present, which can be described as stabilization of a microdroplet of I by putting A at the interface. The structure gradually changes from part a to part b of Figure 2 as more I is added to the system. Interaction between Solvent and Solute. In this section we address the effect of the incompatibility between solute and solvent by varying IS and BI equally from 0.1 to 0.8. The rest of the nonzero interaction parameters, AB and AS, are kept at 0.45, so that IS/AS changes from less than one to greater than one. The volume fraction of polymer, VFABA, is examined at four different values, 0.02, 0.03, 0.04, and 0.055. The volume fraction of small molecules, VFI, varies with VFABA proportionally so that the ratio between solutes and A monomers is constant at 0.96. In our previous work25 we monitored Rg2, the squared radius of gyration of the micellar core comprising both A’s and I’s, with the same parameter set. The Rg2 depended strongly on both IS ) BI and VFABA. At high concentration Rg2 increased remarkedly with IS ) BI, due to an increase in the amount of solute molecules in each micelle, as well as the aggregation number, the average number of A blocks per micelle. Yet this result did not tell us whether the microscopic micellar structure alters or not. To answer this question it is helpful to compare the end-to-end distance of the A blocks in micelles, Re, with the distance of the NAth bead to the micellar center, rNA (Figure 8), as a function of IS ) BI. The values of Re are nearly constant regardless of the concentration of polymer or the strength of solute-solvent interaction. Different data points/curves for rNA are for different volume fractions of copolymers. Comparing Figure 8 with Figure 5 in ref 25, it is noted that rNA resembles the behavior of Rg2 grossly, yet rNA is consistently larger than Rg at every point, (28) Linse, P.; Malmsten, M. Macromolecules 1992, 25, 5434.

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indicating that A-B joints are situated at the surface of each core structure. By increasing either VFABA or IS ) BI, A-B joints become more distant from the center, as a result of the swelling of the cores by solubilizates, as revealed by Figure 11 of ref 24. The solubilization capacity, as defined, measures the volume fraction of I’s in micellar cores on the basis of A and I.24 Its continuous increase with increasing VFABA and IS ) BI shows that the cores are composed of more I’s rather than A segments, due to the accumulation of I’s. If at a certain point rNA exceeds Re, the system undertakes a transition from the first model to the second and a central pool of I starts to form. This transition occurs at the points in Figure 8 where the curves cross the horizontal line representing the end-to-end distance of the A block, Re. The systems for which rNA lies under Re mimic the first model, while those above this line resemble the second model. The slopes of the curves in Figure 8 increase strongly when the transition occurs. This observation has been interpreted as implying different pathways of micellar growth.24 In summary, besides the addition of solubilizates, a higher incompatibility between solvent and solubilizates also promotes the switching from the first model to the second. The higher the polymer concentration, the lower value of IS ) BI at the transition. Adsorption Isotherms. An adsorption isotherm is an expression that gives the fraction θ of a surface that is covered by adsorbed molecules in equilibrium at constant temperature as a function of pressure or concentration. The first quantitative theory of the adsorption was given in 1916 by Langmuir, who based his model on the following assumptions: (1) The uniform surface contains a fixed number of adsorption sites. Each site can hold one adsorbed molecule. (2) There is only interaction between adsorbed molecules and the surface but no interaction between adsorbed molecules on different sites. The chance that a molecule condenses at an unoccupied site or leaves an occupied site does not depend on whether neighboring sites are occupied. (3) The heat of adsorption is the same for all the sites and does not depend on the fraction covered θ. At equilibrium the adsorption isotherm is obtained by equating the rate of desorption and the rate of adsorption. The fraction of surface covered θ is expressed by

1 1 )1+ θ bP

(2)

where P is the gas pressure and the adsorption coefficient b ) ka/kd is the ratio of the adsorption constant to the desorption constant. The Langmuir adsorption equation applies to both chemisorption and physisorption. It works remarkedly well under most circumstances considering it is based on such a simple model. Although the adsorption into the irregular micellar structure is far more complex than that onto a uniform surface, it is worth mentioning that our lattice model meets the second and third assumptions since II ) 0 and the rest of the interaction parameters are also kept constant for a certain ensemble. Furthermore, no correlation between neighboring sites was taken into account. It is of particular interest to probe the applicability of Langmuir adsorption theory to the solubilization in macromolecular surfactants. In a series of simulations where the total concentration of I is the only variable, θ is defined here as being proportional to the volume fraction of solubilizates ad-

Figure 9. Langmuir adsorption isotherm (A15B10A15, AB ) AS ) S ) BI ) 0.45).

Figure 10. Langmuir adsorption isotherm (A15B10A15, AB ) AS ) IS ) BI ) 1.0).

sorbed in the micelles φadsb, θ ∝ φadsb. Meanwhile applying the ideal gas approximation we have P ∝ φfree, where φfree is the volume fraction of the solubilizates dissolved in the solution. Thus the Langmuir adsorption equation can be rewritten as

1 1 )C+ φadsb b1φfree

(3)

where C and b1 are constants. If the adsorption process of the small molecules into the micelles can be described by Langmuir adsorption, we should obtain a straight line when plotting 1/φadsb vs 1/φfree. Figures 9 and 10 show the adsorption isotherm at different interaction parameters, specifically AB ) AS ) IS ) BI ) 0.45 and 1.0, and at different polymer volume fractions. The absorption is strengthened by increasing the polymer concentration and by increasing the hostility between different species as well. When the interaction parameter is 0.45 (Figure 9), approximately straight lines are obtained at all three concentrations. In the presence of minor enthalpic penalties, relatively small aggregates come into being24 and solubilization is comparatively weak. As far as the system observes the first model (Figure 2a), Langmuir adsorption is a good approximation. However, as the interaction parameter is raised to 1.0 (Figure 10), the data follow the linear relationship only at the low φree regime. As the solutes become more concentrated and solubilization gets stronger, the system may change into the second model (Figure 2b). At this stage the first assumption made by Langmuir is no longer valid. Rather than adsorbed by a second agent, the adsorbant condenses

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into droplets where A plays the role of a stabilizing medium. Hence significant deviation from straight lines is evident. An exhaustive investigation29 confirms this conjecture. With application of various approaches demonstrated in this paper, it is seen that accompanying the deviation from the Langmuir adsorption under certain parametric conditions, the system indeed undergoes a transition from the first model to the second one. If we try to make an adsorption isotherm for the strong solubilization discussed in previous sections (conditions for Figures 3-5, 7), the data points are not even close to a straight line. In summary, the Langmuir isotherm provides a fairly good description when the solubilization is comparatively weak and the microstructure can be represented by the first model. In response to the enhancement of enthalpy over entropy, adoption of the structure described by the second model violates Langmuir’s first assumption, and the Langmuir adsorption isotherm becomes a poor description of the system. (29) Xing, L. Ph.D. Thesis, University of Akron, Akron, OH, 1996.

Xing and Mattice

Conclusions A controversy over the validity of two models describing the structure of micelles self-assembled from triblock copolymer and small solubilizates in a selective solvent has been partially resolved. Under the conditions of weak solubilization, the model that depicts a uniform dissolution of solubilizates in a micellar core is the preferred picture and the simple Langmuir adsorption isotherm is obeyed closely. Entering the regime of strong solubilization, a transition of micellar structure into droplet microemulsion is observed and severe violation of the Langmuir isotherm occurs. A two-step solubilization mechanism is therefore proposed to characterize the initial and subsequent accumulation of the insoluble low-mass molecules. Acknowledgment. We thank Professor K. Binder, Dr. W. Paul, and Dr. G. Tanaka for their helpful suggestions about this work. This research was supported by National Science Foundation Grants DMR 9523278 and INT 9421585. LA980104B