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Langmuir 1992,8, 456-459
Intraparticle Distribution Functions for a Micelle Formed by a Small Symmetric Triblock Copolymer in a Poor Solvent for the Terminal Blocks Klein Rodrigues and Wayne L. Mattice’ Institute of Polymer Science, The University of Akron, Akron, Ohio 44325-3909 Received July 1, 1991. I n Final Form: November 8, 1991 The micelleformedby small symmetrictriblock copolymerswith stickyterminal blocks has been simulated on a cubic lattice. The system contains 20 chains, A ~ B ~ o at A ~a ,volume fraction of 0.0376. The chains are non-self-intersecting. The micelle forms when the pairwise interaction of nonbonded beads in the insoluble blocks is 0.5kT when either a bead from a soluble block or a void (solvent)is a nearest neighbor. The micelle has a rudimentary internal core that is dense-packed beads of A, a very diffuse interface, and a less expanded corona than a diblock copolymer micelle. Both solvent and B beads penetrate further into the core of the micelle of a triblock copolymer than into the core of a micelle of a diblock copolymer. Introduction Diblock copolymers, abbreviated here as A-B, can easily form micelles in dilute solution in a medium that is a poor solvent for A and a good solvent for B. The experimental studies of these micelles, which are far too numerous to be cited completely, date back over three decades.lV2 The micelles have also been studied by theoretical methods3 and by computer simulation^.^.^ The behavior of the system becomes more intriguing when the monomer units in the A-B diblock copolymer are rearranged to produce an A-B-A triblock copolymer. Experiments provide evidence for the aggregation of some triblock copolymer^,^^ but not others,1e12 in media that are good solvents for the internal block and poor solvents for the terminal blocks. Intramolecular association is seen in some of the systems that do not exhibit aggregation.lOJ1 Theoretical arguments have been advanced forg and against12the formation of micelles by symmetric triblock copolymers. Computer simulations support the formation of independent micelles under some conditions, but more complicated structures, consisting of clusters or networks of interconnected micelles, are seen under other conditions. l3 Here we examine the internal structure of the independent micelles observed in the computer simulation of dilute solutions of the symmetric A-B-A triblock copolymers. Specifically, we describe the sharpness of the interface between the core and the corona, the density and internal structure of the core, the size of the corona, and the distribution of solvent within the micelle. A (1) Gallot, Y.; Leng, M.; Benoit, H.; Rempp, P. J. Chem. Phys. 1962, 59,1093. (2)Gallot, Y.;Franta, E.; Rempp, P.; Benoit, H. J.Polym. Sci., Part C: Polym. Symp. 1964,4, Part C, 473. (3)Halperin, A. Macromolecules 1987, 20, 2943. (4)Rodrigues, K.; Mattice, W. L. J. Chem. Phys. 1991, 94, 761. (5) Rodrigues, K.; Mattice, W. L. J. Chem. Phys. 1991, 95,5341. (6)Krause, W.J.Phys. Chem. 1964, 68, 1948. (7)Kotaka, T.; Tanaka, T.; Hattori, M.; Inagaki, H. Macromolecules 1978, 11, 138. (8) Plestil, J.; Hlavata,D.; Hrouz, J.; Tuzar, Z. Polymer 1990,31,2112. (9)Balsara, N.P.;Tirrell, M.; Lodge, T. P. Macromolecules 1991,24, 1975. (10)Tanaka, T.; Kotaka, T.; Inagaki, H. Polym. J. (Tokyo) 1972, 3, 327. (11)Tanaka, T.; Kotaka, T.; Inagaki, H. Polym. J. (Tokyo) 1972, 3, 338. (12) Tang, W. T.; Hadziioannou, C.;Cotts, P. M.; Smith, B. A.; Frank, C. W. Polym. Prepr. (Am. Chem. Soc., Diu. Polym. Chem.) 1986,27 (3, 107. (13)Rodrigues, K.; Mattice, W. L. Polym. Bull. 1991, 25, 239.
0743-746319212408-0456$03.0OlO
comparison is made with the recent results? obtained by identical methods, for other micelles of the same mass and ratio of A to B, but composed of chains of A-B diblock copolymers instead of A-B-A triblock copolymers. Method The simulation was performed on a 22 X 22 X 22 cubic lattice with periodic boundary conditions, as described previously for diblock copolymers.* The lattice contained 20 chains, each comprised of two terminal blocks with five A beads each, and an internal block of ten B beads. All bonds are of unit length. The voids are considered to be occupied by solvent, S. The volume fraction of the copolymer is 0.0376. No lattice site can be doubly occupied by copolymer. Nonbonded nearest neighborsare subject to the interactions C’AB = €’AS = 0.5, where C’XY = 6 x y J k T . Under these conditions the 20 chainsformasingle micelle.’3 The system also forms a singlemicelle if the beads in each chain are rearranged to produce an A-B diblock cop~lymer.~.~
Results and Discussion A. Snapshot of the Micelle. The four panels of Figure 1 depict a snapshot of the micelle composed of 20 chains of A-B-A. All bonds are depicted in panel a. This panel shows that the micelle has a compact structure. Panel b depicts only the bonds, 160 in number, between two beads of A. Most of these bonds are located in the core of the micelle, but a few of the A blocks are on the periphery of the micelle, where they are exposed to solvent. In many of the snapshots, two to four of the A blocks are immersed in the s01vent.l~Panel c depicts only those bonds, 180 in number, between two beads of B. These bonds prefer a location in the corona of the micelle. Panel d depicts only the bonds between A and B. The distribution of these 40 bonds is determined by the sharpness of the interface. The distribution functions described below are averaged over 75 independent snapshots equivalent to those depicted in Figure l . The origin for all distribution functions is the center of mass of the entire micelle. In the averaging, we exclude a small minority of the snapshots (on the order of 1 0 % ) which show a structure different from a single micelle of 20 chains. Most of the snapshots that are ignored contain a micelle of 19 chains and another chain that is separate from the micelle.l3 B. The Interface. We start with the distribution function for the A beads that are bonded to a B bead. These beads lie at the interface between two blocks. The points that define the distribution function are calculated for spherical shells at intervals of 1 for r2,where r denotes 0 1992 American Chemical Society
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Table I. Mean Squared Radii of Gyration (in Units of a2) for Different Portions of the Micelles (IQ = 13.2) portion of the micelle all insoluble blocks free ends of the insoluble blocks junctions entire micelle all soluble blocks free ends of soluble blocks
diblock6 0.68 0.82
0.92 1.43 2.18 3.07
triblock 0.89 0.96 0.99 1.21 1.53 a
The soluble block has no free ends. Table 11. Other Features of the Micelles ____ featurea diblock triblock largest r2 for which V A = 1 0.5b2 O.lb2 largest 1.2 for which U A > U B 1.14b2 0.98b2 r2 with largest U B ~ 1.4b2 L l b 2 largest f for which U B > ( U B ) 4b2 3 b 2 U A and UB denote the volume fractions of A and B, respectively, in their spherical shells centered on r. The largest U B is 0.27 for the diblock, and 0.39 for the triblock. ~~~
Figure 1. Projectionsin twodimensionsof a snapshotof a typical configuration of the micelle formed by 20 symmetric triblock on a cubic lattice. The four panels depict copolymers, A&&, (a) all 380 bonds in the micelle, (b) all 160 bonds in the insoluble blocks (all A-A bonds), (c) all 180 bonds in the soluble blocks (all B-B bonds),and (d) all 40 bonds at the junctions (allA-B bonds).
*
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the distance from the center of mass. The result, averaged over 75 independent snapshots, or 3000 junction points, is depicted as the filled squares in Figure 2. It is compared with the distribution function (open circles) reported previously for the A beads at the A-B junctions in the micelle formed by an A-B diblock ~opolymer.~ Both micelles contain 200 beads of A and 200 beads of B distributed over 20 identical chains, and both micelles are subject to the same pairwise interaction energies. The micelles differ only with regard to whether the component chains have their monomer units arranged as A5B1d6 or AloBlo. The micelle formed from the triblock copolymers has a broader distribution for the junction points than does the micelle formed from the diblock copolymers. Evidence for its greater breadth is provided by the lower maximum in the vicinity of r2 = 10, and the more prominent tail a t 20 < r2 < 94. The mean square radius of gyration of the junction points, denoted by (rjz), is 13.1 for the micelle formed by the triblock copolymers, and 12.1 for the mi-
celle formed by the diblock copolymers. The origin of the long tail in the distribution function for the junction points in the micelle formed by the triblock copolymers arises from the conformational entropy of the internal soluble block. Ita conformational entropy must be reduced if the B block is to be located in the corona, and place the A blocks at both ends in the core. Sometimes a B block will resist this reduction in its conformational entropy by leaving one of the A blocks in the solution as a dangling end, even though this situation is energeticallyunfavorable for the A block. Several examples are depicted in panel b of Figure 1. The junction points in these dangling ends produce the long tail in the distribution function in Figure 2. With larger values of C’AB and C’AS, the dangling ends promote interaction between micelles, with the formation of ~1usters.l~ This phenomenon at the molecular level would account for the use of A-B-A triblock copolymers as thickeners. In order to escape from the lattice spacing as the unit for expressing (rj2),it is useful to focus on the geometry of the core in a hypothetical micelle that has 200 A beads arranged in a close-packed sphere. The squared radius of that sphere is obtained as
rh2 = ( 3 ~ / 4 a ) ~ / ~ (1) where V is 200 in units of lattice spacing. Thus, the hypothetical close-packed sphere has a squared radius of 13.2. If the diblock and triblock copolymers formed micelles with a close-packed spherical core of A, both distribution functions in Figure 2 would be 6 functions located at r2 = 13.2, because all junction points would be on the surface of the sphere. Neither micelle conforms to this simplistic model, and the deviations from the model are more pronounced for the triblock copolymers than for the diblock copolymers. Nevertheless, the values of ( rj2) are quite close to ~ ~ as2summarized , in Table I. In the following text, we will refer to that portion of the micelle for which r2 < ( rj2) as the ”core”. The boundary between the core and the corona is diffuse, as shown by the distribution functions in Figure 2. C. The Entire Micelle. Figure 3 depicts the distribution functions for all beads in the micelles. These distribution functions can be divided into three parts. In , distribution the internal portion of the core, r2< 0 . 7 b 2 the functions for the diblock and triblock copolymer micelles are indistinguishable. In the outer core and internal portion of the corona, 0.7rh2 < r2 < 2 ~ 2the , distribution function for the triblock copolymer micelle is larger than
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Figure 3. Distribution functions for all beads in the micelle formed by the A-B-A triblock copolymers (filled squares) and A-B diblock copolymers (open circles). 0.08
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'Hes Figure 4. Distribution functions for the insoluble A blocks in the A-B-A triblock copolymers (filledsquares)and A-B diblock copolymers (open circles). the distribution function for the diblock copolymer micelle. The opposite relationship is seen in the outer corona, r2> 2 h 2 . The narrower distribution function for the triblock copolymer micelle causes it to have a mean squared radius of gyration that is 15% smaller than the diblock copolymer micelle. The difference can be attributed to the increased probability that one end of the B block will be far from the interface when that block is in A-B instead of A-B-A. D. The InsolubleBlocks. The distribution functions for the insoluble A beads are depicted in Figure 4. They cross in the vicinity of rh2. The core of the diblock copolymer micelle is richer in A than the core of the triblock copolymer micelle, and the corona of the triblock copolymer micelle is richer in A than the corona of the diblock copolymer micelle. The molecular origin of the difference in the distribution for A lies in the sacrifice in conformational entropy for the central B block in the triblock copolymer if it is to place both of ita terminal A blocks in the core.
Figure 5. Distribution functions for the soluble B blocks in the A-B-A triblock copolymers (filled squares) and A-B diblock copolymers (open circles).
The two distribution functions in Figure 4 become identical very close to the center of mass, where the volume fraction of A is very close to 1in both micelles. The portion of the core in which the volume fraction of A remains very near 1 is larger in the micelles formed by the diblock copolymers than in those formed by the triblock copolymers. In the former case it extends out to about 0.5h2, but in the latter case it reaches only to 0.1b2. The difference in the sizes of the close-packed internal cores of A is directly related to the distribution functions for the soluble B beads, which follow in the next paragraph. E. The Soluble Blocks. The distribution functions for the soluble B beads in the micelles cross at about 2 h 2 , A t smaller r2 there is a higher probability for B beads in the micelle of the triblock copolymer than in the micelle of the diblock copolymer. This difference extends to the center of mass of the micelle. The B beads can penetrate much closer to the center of mass in the triblock copolymer than in the diblock copolymer. Consequently, the extent of the internal close-packed core of A is much smaller in the triblock system than in the diblock system. The mean squared radius of gyration of the soluble B beads in the triblock copolymer micelle is about 30% smaller than its value for the diblock copolymer micelle. The ends of the B blocks in the triblock copolymer have a squared radius of gyration very similar to (rj2),and this number is only about 2/3 as large as the squared radius of gyration for all B beads. Consequently, the ends of the B block tend to be closer to the center of mass of the micelle than do the other beads in the B block. This situation obviously arises from the tendency for the two A blocks to be in the core. F. The Free Ends of the Insoluble Blocks. Figure 6 depicts the distribution functions for the A beads a t the free ends of the A blocks. In comparing the two types of micelles, it can be seen that the free ends of the A blocks in the triblock system have a greater probability of being found near the center of mass than was the case for the diblock system, but they also have a greater probability of being found far out in the corona. Perhaps more attention should be given to the comparisons for the distribution functions for all A beads (Figure 41, the A beads at the A-B functions (Figure 2), and the A beads at the free ends of the A blocks (Figure
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Figure 7. Fractionof the sites occupied by solvent,& a function of the square of the distance from the center of mass of the micelle formed by the triblock copolymers.
6). These three distribution functions are remarkably similar for the micelle formed by the triblock copolymers. The mean square radii of gyration extracted from these three distribution functions (Table I) fall in the narrow range of 0.89-0.99b2. A much greater discrimination between the internal and terminal beads in the A block is seen in the micelles formed from the diblock copolymers. There the three mean square radii of gyration have the broader range 0.6&0.92h2. G. Penetration of Solvent into the Micelle. Figure 7 depicts the ability of solvent (voids) to penetrate into the micelle. The qualitative appearance of this curve is very similar to the one obtained for the diblock copolymer m i ~ e l l e .The ~ diblock copolymer micelle has slightlymore solvent in the region 0.75h2< r2 < 2.3h2, and the triblock copolymer micelle has more solvent outside this range. The triblock copolymer micelle has more solvent at very large r2 because its corona is less extensive, and it has more solvent at very small r2 because the close-packed internal core of A is not as well organized in the triblock copolymer as it is in the diblock copolymer.
Conclusion
A simplisticmodel for a block copolymer micelle w u m e s a dense close-packed sphericalcore of the insolubleblocks, junction points at the surface of the sphere, and a corona of the solubleblocks that extends outward from the surface of the sphere. The simplistic model provides a good estimate of the mean square radius of gyration of the junctions in the micelles formed by small diblock and triblock copolymers. Ita failures are in a misleading picture of the diffuse nature of the interface, an overestimate of the size of the dense-packed central core of A, failure to recognize the penetration of the solvent and soluble block into the core, and failure to recognize the extension of the insoluble beads into the corona. The simplistic model is a poorer descriptionof the triblock copolymer micellethan of the diblock copolymer micelle. Acknowledgment. This research was supported by Grant No. DMR 90-14502 from the National Science Foundation.
