Formation of Surface Micelles from Adsorbed Asymmetric Block

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Formation of Surface Micelles from Adsorbed Asymmetric Block Copolymers: A Monte Carlo Study Andrey Milchev† and Kurt Binder* Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria, and Institut fu¨ r Physik, Johannes-Gutenberg-Universita¨ t, D-55099 Mainz, Germany Received October 6, 1998. In Final Form: December 22, 1998 The properties of surface micelles formed from flexible short block copolymers of composition AfB1-f with f ) 1/4 and chain lengths of N ) 8 and N ) 16 effective monomers are obtained from Monte Carlo simulation of a coarse-grained bead spring model. We consider the case where the surface has a short-range attraction to the A monomers but is repulsive for the B monomers. Depending on the surface coverage and the strength EAA of the attractive energy between the A monomers, the adsorbed polymers may form “mushrooms”, or a polymer brush, or surface micelles. We show that a rather well-defined critical micelle concentration occurs, similar to that for micelle formation in bulk solutions, and we study the size distribution of the micelles as well as their internal structure. Large micelles exhibit a very dense solid core and are essentially immobile. Equilibration then is still possible via the exchange of single mobile chains between micelles.

1. Introduction The self-assembly of surfactant molecules into micelles in solutions in the bulk is a well-known process which has received long-standing attention.1-4 In contrast, much less is known on the corresponding self-assembly of surfactant molecules into surface micelles at interfaces. For some time surface micelles have been discussed in the context of Langmuir monolayers at the water-air interface,5-7 to explain the nonhorizontal part of gas-liquid spreading pressure versus area isotherms. Only recently great interest arose in surface micelles at solid substrates, because of the observation that these surface micelles may arrange in quasi-regular patterns.8 Such ultrathin films with a nearly regular structure on the nanoscale may find various applications, and thus a better physical understanding of the conditions under which they form is desirable, as well as a detailed knowledge of their properties. Computer simulation of simple models for these systems should be useful, since the simulations allow the variation of parameters that cannot be as easily controlled (and often are unknown) in experiment, and yield a very complete and detailed information on the nanoscopic scale.9 In fact, recent computer simulations of micelle formation of short block * To whom correspondence should be addressed. † Present and permanent address: Bulgarian Academy of Sciences. (1) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley: New York, 1980. (2) Lindman, B.; Wennerstro¨m, H. Topics in Current Chemistry; Springer: Berlin, 1980; Vol. 87, p 1. (3) Mittal, K. L., Lindman, B., Eds.; Surfactants in Solution; Plenum: New York, 1984. (4) Degiorgio, V., Corti, M., Eds.; Physics of Amphiphiles: Micelles, Vesicles, and Microemulsions; North-Holland: Amsterdam, 1985. (5) Ishraelashvili, J. Langmuir 1994, 10, 3774-3781. (6) Faineman, V. B.; Vollhardt, D.; Melzer, V. J. Phys. Chem. 1996, 100, 15478-15482. (7) Kaganer, V.; Mo¨hwald, H.; Dutta, P. Rev. Mod. Phys. 1999, in press. (8) Spatz, J. P.; Sheiko, S.; Mo¨ller, M. Adv. Mater. 1996, 8, 513; Macromolecules 1996, 29, 3220-3226; Spatz, J. P.; Mo¨ller, M.; Noeske, M.; Behm, R. J.; Pietralla, M. Macromolecules 1997, 30, 3874-3880; Siqueira, D. F.; Kohler, K.; Stamm, M. Langmuir 1995, 11, 3092. (9) Binder, K., Ed.; Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Oxford University Press: New York, 1995.

copolymers in dilute solution in the bulk have yielded a very comprehensive and useful picture.10-13 In the present paper, we hence study the self-assembly of short block copolymers, where one block is strongly adsorbed at a hard wall. Section 2 briefly describes our model and discusses methodological aspects of the simulations, while section 3 presents our results for the critical micelle concentration (cmc) of the adsorbed block copolymers. Section 4 discusses the internal structure of the surface micelles, presenting results for density distributions of A monomers and B monomers in radial direction both parallel to the surface and perpendicular to it. Section 5 comments on the dynamic behavior of our model, while section 6 summarizes some conclusions. 2. Model and Simulation Technique For the sake of computational efficiency, we dispose here of any chemical detail of the simulated polymer chains, addressing the generic features of surface micelle formation only. Thus, we imagine that several chemical bonds are integrated into one “effective” bond,9 which then is represented by the FENE (finitely extensible nonlinear elastic) potential,14 where the length l of an effective bond is restricted to the interval lmin < l < lmax:

UFENE (l) ) 1 - K(lmax - l0)2 ln[1 - (l - l0)2/(lmax - l0)2] (1) 2 The minimum of this potential occurs for l ) l0, UFENE(l0) ) 0, and near l0 it is harmonic, UFENE(l0) ≈ (1/2)K(l - l0)2. But this potential diverges logarithmically both when l f (10) v. Gottberg, F. K.; Smith, K. A.; Hatton, T. A. J. Chem. Phys. 1997, 106, 9850-9857. (11) Nelson, P. H.; Rutledge, G. C.; Hatton, T. A. J. Chem. Phys. 1997, 107, 10777-10781. (12) Viduna, D.; Milchev, A.; Binder, K. Macromol. Theory Simul. 1998, 7, 649-658. (13) Bhattacharya, A. Ionic Micelles and Co-operative Self-Assembly; preprint, 1998. (14) Milchev, A.; Paul, W.; Binder, K. J. Chem. Phys. 1993, 99, 47864798; Milchev, A.; Binder, K. Macromol. Theory Simul. 1994, 3, 915930.

10.1021/la981387a CCC: $18.00 © 1999 American Chemical Society Published on Web 04/02/1999

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lmin ) 2l0 - lmax and when l f lmax. Note that we choose the parameters K, lmax, and l0 the same for all effective bonds, irrespective of whether we deal with a bond connecting two effective monomers of type A, or type B, or the junction bond connecting A and B. We choose our unit of length such that lmax ) 1 and the other parameters are then chosen as

l0 ) 0.7, K/kBT ) 40

(2)

as in previous work on the adsorption of homopolymers.15-17 Between any nonbonded pair of effective monomers, irrespective of whether we deal with an intrachain or an interchain interaction, we use Morse potentials:

Uij(r) ) Eij{exp[-2R(r - rmin)] 2 exp[-R(r - rmin)]} + C (3) The strength Eij of these potentials depends on the type of monomers:

3 e EAA e 4.3, EAB ) EBB ) 1.0

(4)

choosing units such that Boltzmann’s constant kB ) 1 and the absolute temperature T ) 1. The parameter R which controls the range of the potential is chosen to be rather large, R ) 24, so we have an extremely short-range potential, and R is also chosen independent of the type of pairs, as is the equilibrium distance rmin, rmin ) 0.8. We furthermore put Uij(r g rcutoff) ) 0, with rcutoff ) 1. The constant C in eq 3 is chosen such that U is regular at r ) rcutoff. This choice of parameters has the advantage that a fast link-cell algorithm with cell size unity can be used.14 If A monomers that belong to different chains are at a distance r < 1, within the range of the attractive potential eqs 3 and 4, we count the respective chains as part of the same micelle. Because of the coarse-grained nature of this model (Figure 1), we do not employ any torsional potentials and bond angle potentials, respectively. We choose block copolymers of composition f ) NA/N ) 1/4, treating two choices of the chain length N ) NA + NB, namely, N ) 8 (Figure 1a) and N ) 16, respectively. Note that no solvent molecules are included explicitly; their effect is thought to be included implicitly via the choice of the effective potentials Uij(r). The advantage of these drastic simplifications is that we can deal with a reasonably large number of effective monomers (typically of the order of several thousand). If we chose also EAA ) 1, then the model would reduce precisely to the homopolymer model studied previously.14-17 Note that for this model the θ temperature is θ ≈ 0.62; that is, for T ) 1 one is in the good solvent regime. Thus, also in the present case (eq 4) there is no tendency between B-B pairs or A-B pairs to attract each other. The potential between these pairs is still effectively repulsive, while the situation is clearly different with respect to the A-A pairs: if we considered homopolymers of type A here, we would be in the bad solvent regime (T ) 1 is a temperature below the θ temperature, θAA ) EAAθ ≈ 0.62EAA throughout). Since only A monomers effectively attract each other, while A-B pairs and B-B pairs repel each other, we define the case of a micelle by the A monomers that it contains, while the B monomers form the corona. (15) Milchev, A.; Binder, K. Macromolecules 1996, 29, 343-354. (16) Binder, K.; Milchev, A.; Baschnagel, J. Annu. Rev. Mater. Sci. 1996, 26, 107-134. (17) Pandey, R. B.; Milchev, A.; Binder, K. Macromolecules 1997, 30, 1194-1204.

Figure 1. (a) Schematic description of the coarse-grained beadspring model of adsorbed flexible block copolymers, choosing NA ) 2 effective monomers of type A and NB ) 6 effective monomers of type B. Nearest neighbor monomers interact with the anharmonic spring potential UFENE(l) [cf. text] while between any nonbonded pair of monomers we assume short-range Morsetype potentials Uij(r), where indices i and j stand for A or B, respectively. A monomers are bound with a potential UA(z) to the attractive wall situated in the plane z ) 0. (b) Geometry chosen for the simulation box of linear dimensions L × L × D. Periodic boundary conditions are chosen along the x- and y-directions parallel to the confining walls.

The geometry of the simulation box (Figure 1b) was chosen L × L × D, with two hard unpenetrable walls of area L × L at z ) 0 and at z ) D, respectively, choosing periodic boundary conditions in the lateral x,y-directions. At the lower wall at z ) 0, we use an attractive potential of range unity that acts on the A monomers only:

Ui(z) ) Ei[exp(-2Rz) - 2 exp(-Rz)], i ) (A, B), EA ) 6, EB ) 0 (5) With this choice the density of monomers far from the attractive wall is very small, and thus the precise location of the repulsive wall (i.e., the choice of the slab thickness D) has a negligible influence on the simulation results only. Of course, without this confining repulsive wall a chain once released from the attractive wall could escape off toward z f ∞ while with the present geometry it will diffuse in the simulation box until it gets readsorbed at the attractive wall. So the chosen geometry ensures that a stable thermodynamic equilibrium can in principle be achieved. Again, we note that eq 5 is meant only as a qualitative description of strong (chemical) forces between the attractive wall and the monomerssthe weak longrange van der Waals-type forces (that decay ∝ z-3) are disregarded here throughout. Typical linear dimensions chosen are then 32 × 32 × 8 and 64 × 64 × 8 (for N ) 8) and 32 × 32 × 16 (for N ) 16, respectively). For these choices, systematic finite size

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Figure 2. Snapshots of equilibrated chain configurations at a (normalized) monomer density φ ) 0.109 375, choosing box linear dimensions L ) 64 and D ) 8, for N ) 8 and choice of parameters EAA ) 3 (a) and EAA ) 4 (b). Note that the left and right corners of the adsorbing plane in the foreground are partially cut off, for the sake of a better visibility of the remaining configuration.

effects have been found to be smaller than the typical statistical error, and hence also need not be considered. As a simulation algorithm, we mostly employ a “dynamic” Monte Carlo algorithm,9,14 choosing a particular

monomer of a randomly selected chain at random for a trial displacement. The new position is drawn randomly from a cube of volume unity, the cube being centered at the old position. Typically, times of the order of 12 million

Monte Carlo Study of Asymmetric Block Copolymers

Monte Carlo steps (MCS) per monomer are used for equilibration and 42 million MCS for taking averages. One should note, however, that, in the case of large micelles where a rather dense packing of effective monomers of type A is achieved, the acceptance rate of the Monte Carlo trial moves is extremely low: effectively, these micelles are frozen-in and immobile. However, equilibration is still possible because of “evaporation” of single chains from the boundary region of a micelle; these single chains can freely move along the attractive surface via surface diffusion, until they condense on other micelles. We have checked that under the conditions studied still some single chains were always present and thus the system was not completely frozen. Of course, similar problems were reported for simulations of micellar solutions in the bulk.10-12 As an alternative to this dynamic Monte Carlo algorithm of a Rouse-model-like nature, also “slithering snake”9 and “configurational bias”9 algorithms were tried, but found to offer only little advantage here. In section 5, where we describe some data for dynamical properties of micelles, evidence will be given that for most parameters of interest correlation functions decay on time scales of 1-10 million MCS. Hence, we have established that the present statistical effort was enough to achieve meaningful results. To give a visual impression of the simulated system, Figure 2 presents two snapshot pictures of 448 chains in a box of sizes L ) 64, D ) 8, one for EAA ) 3 (which leads to a situation where mostly “mushrooms” occur at the surface, and about 28% of the chains are not adsorbed on the surface, but move freely in the bulk), and one for EAA ) 4 (where many micelles are formed, which are mostly adsorbed at the attractive wall, only 7.7% of the chains being not adsorbed). At this point, we emphasize that our choice of parameters (very short-chain lengths N, fixed f ) 1/4, good solvent conditions for B monomers, bad solvent conditions for A monomers, and only the latter are attracted to the wall) provides only one facet of self-assembly, namely, formation of surface micelles, out of a wealth of complex structure formation phenomena.18 If one chooses bad solvent conditions for both monomers,18 “onion” and “garlic”-type structures as well as polymer brushes are formed. On the basis of scaling considerations, Zhulina et al.18 proposed various diagrams of states with different structures of the grafted diblocks for various choices of parameters and obtained density profiles of the various types of aggregates from self-consistent field calculations. While for the case of selective solvents the case of bad solvent for A and θ solvent for B was studied, the case of the present paper (bad solvent for A, good solvent for B) has not been studied. Finally, we draw attention to the fact that related phenomena could occur also for other polymer architectures (e.g., graft copolymers19 instead of block copolymers). 3. Critical Micelle Concentration (cmc) and Micellar Size Distribution Figure 3 shows the concentration of single chains, φ1, plotted versus the total density φ of monomers, for two typical systems (N ) 8 and EAA ) 4 as well as N ) 16 and EAA ) 3, respectively). The initial variation at low density is the ideal gas behavior, indicated by the straight line. The formation of micelles is evident from the maximum of these curves. The slight decrease of φ1 with φ after the (18) Zhulina, E. B.; Singh, C.; Balazs, A. C. Macromolecules 1996, 29, 6338-6348; 29, 8254-8259. (19) Balazs, A. C.; Gersappe, D.; Israels, R.; Fasolba, M. Macromol. Theory Simul. 1995, 4, 585-612.

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Figure 3. Monomer density in single chains (φ1) plotted vs total monomer density for N ) 8 and EAA ) 4 (full dots) and for N ) 16 and EAA ) 3 (squares). The straight line shows the ideal gas law. For N ) 8 the linear dimensions chosen are L ) 64 and D ) 8, and for N ) 16, L ) 32 and D ) 16, respectively.

Figure 4. Monomer density in single chains (φ1) of length N ) 8 plotted vs EAA, for a system at density φ ) 0.094 (384 chains in a box with L ) 64 and D ) 8) and for a system at density φ ) 0.125 (128 chains in a box with L ) 32 and D ) 8).

maximum has occurred can be attributed to the “excluded volume” interaction between micelles.10-12 Note that for N ) 16 and φ g 0.17 we find that φ1 increases again: this behavior is due to the fact that then the surface gets “saturated” with surface micelles. Additional chains do not all get adsorbed on the attractive wall, but rather desorb in the bulk of the slab. For this choice of EAA one is beyond the cmc at the surface, but has not exceeded the cmc of the bulk for the (remaining) small density of chains that is not adsorbed on the surface but is in the bulk of the slab. Therefore, φ1 must increase with φ. Figure 4 studies the variation of φ1 with EAA, choosing two different box lateral dimensions to demonstrate that there are no visible finite size effects. It is seen that for EAA g 3.8 (E crit AA ) one has a simple exponential behavior, φ1 ∝ exp(-3.8EAA/kBT). Assuming from Figure 3 that the cmc can be defined as cmc ) φ1 (in the flat region, ignoring the systematic decrease with φ due to the excluded vol-

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Figure 5. Plot of average aggregation number MW versus EAA, for the chain length N ) 8 and the two systems described in Figure 4.

Figure 6. Micellar size distribution ns plotted vs s, for the case N ) 8, φ ) 0.09375, and four different choices of EAA.

ume interactions), one would identify 3.8EAA with the enthalpy of micellization ∆GM, in the regime where φ1 ∝ exp(-3.8EAA/kBT) holds. Hence, one finds the expected simple law

∆GM/kBT ) -ln(cmc),

EAA g 3.8

(6)

Next, we turn to the size distribution ns of the micelles (ns is the number of micelles per unit volume that contain precisely s chains). A good measure of micelle formation is the average aggregation number, or “molecular weight” Mw, of the micelles, defined from the second moment of ns: ∞

Mw )

/

∞

∑ s2ns s)2 ∑ sns... s)2

(7)

For the snapshots shown in Figure 2, Mw ) 2.64 in case (a), where there are hardly any micelles but just single chains and a few dimers, trimers, and so forth, but Mw ) 8.45 in case (b), where micelles were clearly visible. Figure 5 then shows a plot of Mw versus EAA, for the two systems shown in Figure 4. Now, one sees that Mw is systematically higher for the larger value of φ, unlike φ1 where no difference was detectable (Figure 4). The full distribution ns is shown in Figures 6 and 7. It crit is seen that for EAA < E crit AA (E AA ≈ 3.8 for N ) 8 and φ ) 0.125) the size distribution decreases rapidly and monotonically with increasing s. But, for EAA ≈ E crit AA a shoulder in the distribution develops, which grows into a pronounced peak for EAA > E crit AA . This behavior is indicative of the fact that many large (surface) micelles have been formed (the evidence that the micelles are indeed adsorbed and not off the surface in the slab will be described in the next section). A similar change in the shape of the size distribution can also be detected at fixed EAA > E crit AA when φ is varied; see Figure 7. In this way once more a criterion to find the cmc can be established, using the value of φ where a shoulder first appears. We have also analyzed the shape of the micelles studying the eigenvalues of the gyration tensor and found that the shapes are essentially compact (hemispherical). Guided by the experience with micelles in the bulk,12 this finding

Figure 7. Micellar size distribution ns plotted vs s, for different choices of φ, for the case N ) 8 and EAA ) 4 (a) and the case N ) 16 and EAA ) 3 (b).

is expected: cylindrical shapes occur only for Mw g 40, while here Mw e 20 throughout.

Monte Carlo Study of Asymmetric Block Copolymers

Figure 8. Radial density distribution of the micellar core region (A-mers, open symbols) and the micellar corona region (B-mers, full symbols) at constant surface coverage φs ) 0.5 but three different choices of the chain lengths, as indicated in the figure. Data are obtained by averaging over all micelles of sizes s g 3. Note that r is a coordinate in the lateral direction only (r ) (xx2+y2)).

4. Structural Properties of the Surface Micelles Figure 8 gives some averaged information on the density distribution of the surface micelles. It is seen that the density in the core is about an order of magnitude larger than that in the corona. Actually, the density reached in the core is that of a dense melt or even a glassy or crystalline state. This fact is demonstrated in Figure 9, where snapshots of the locations of adsorbed A-mers at various choices of EAA are compared to the corresponding radial distribution functions of A-mers (GAA) and whole micelles (G). While for EAA ) 3.5 there are still many single chains present (NA ) 2 A-mers close together), and small clusters containing s ) 2, 3, 4, ... chains, for EAA g 3.8 the formation of several large micelles can clearly be recognized. The spiky structure of GAA(r) reflects a crystal-like arrangement of the A-mers in the micellar cores, as can be seen rather clearly in the snapshot pictures. At this point, we recall that experimental studies of micelles in bulk solutions have revealed cases with crystalline core, for example, polyethylene-poly(ethylenepropylene) suspended in decane or poly(ethyleneoxide)-polystyrene in cyclopentane.20,21 But, because of the use of a much larger molecular weight the crystalline domains there form thin platelet structures containing folded chains as are typical for semicrystalline polymers. In our model with very short oligomers, where each A block contains only NA ) 2 effective monomers of type A, chain folding is impossible. The only constraint limiting the size and shape of the crystals that form is the B chains attached to the A blocks; therefore, the linear dimension of the “crystals” in the x,y-direction is clearly larger than that in the z-direction, as a closer inspection of the snapshots in Figure 9 shows (the height of the crystals is typically 3-4 A-mers). The crystals, when they have formed, are absolutely immobile on the substrate. Therefore, one obtains also several spikes in the radial distribution function G(r) of the center of (20) Richter, D.; Schneiders, D.; Monkenbusch, M.; Willner, L.; Fetters, L. J.; Huang, J. S.; Lin, M.; Mortensen, K.; Farago, B. Macromolecules 1997, 30, 1053-1068. (21) Lin, E. K.; Gast, A. P. Macromolecules 1996, 29, 4432-4441.

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gravity of the whole micelles. (These spikes are absent in Figure 9b, where the micelles have still fluid cores, and a small lateral mobility of the micelles washes out this special structure of G(r), although GAA(r) already shows structure.) Another interesting feature is the pronounced correlation hole effect22 that is visible in G(r) always; this can be attributed to the excluded volume interaction between micelles that is already effective in the corona regions. It is also interesting to compare the three different radial distribution functions GAA(r), GBB(r), and GAB(r) with each other (Figure 10a). All three correlation functions have a peak near r ≈ 0.7, because of the neighborhood of bonded monomers along a chain. But it is only GAA(r) that has sharp peaks at larger distances as well, reflecting the dense packing of A-mers in a crystalline arrangement. Between r ≈ 2.5 and r ≈ 5.5 the function GAA(r) clearly falls below its asymptote for large distances, GAA(r f ∞) ) 1: this again is a kind of “correlation hole” effect, since also the corona region of a micelle is excluded for the case of other micelles. The distribution function GAB(r), on the other hand, is fairly large at distances from r ) 1.5 to r ) 1.3, because of next-nearest and third-nearest B-mer neighbors along a chain, which must also be geometrically close to the A-mers that belong to the micellar core. It is interesting that this distribution function GAB(r) also has a shallow minimum for 4 e r e 6: these are distances just outside the radius of the corona of the micelle, and because of the excluded volume interaction between micelles as a whole, the density of B-mers is slightly reduced there. In GBB(r), on the other hand, this minimum is washed out because one considers correlations between any B-mer in a micellar corona with any other one. The micellar corona has a very disordered structure, with a low B-mer density corresponding to the semidilute regime of polymer solutions.22 These interpretations are strengthened when one considers the variation of GAA(r) and GBB(r) with the strength of the interaction EAA between the A-mers (Figure 10b,c): Only when EAA exceeds the micelle formation energy ∆GM ≈ 3.8 do sharp features in GAA(r) besides the nearest neighbor peak develop, and the correlation hole for r > 3 becomes deep. At the same time, GBB(r) gets enhanced for 1.5 e r e 5 since the condensation of the A-mers of many chains in the micellar cores creates an enhanced density of B-mers in the region of the corona adjacent to the core as well, for purely geometric reasons. The density distribution of B-mers in a micellar corona is similar to that of a star polymer.23 Near the core semidilute concentrations are reached, while far away from the core the concentration can be rather dilute. Finally, we turn to the density profiles nA(z) and nB(z) in the z-direction perpendicular to the adsorbing wall (Figure 11). It is seen that there is a very sharp peak at z ) 0 (the peak reaches a value of around 50 in Figure 11a; that is, it comes close to a δ function!). This peak represents the first layer of the “crystal” that is tightly bound to the substrate. There is a second sharp peak near z ≈ 0.6, representing the second layer of the crystal. Note that the A-mers in the second layer do not sit exactly on top of the A-mers in the first layer, but rather at shifted positions such as in the stacking of close-packed planes in an fcc or hcp arrangement. If the structure were perfect, we hence would expect the second peak as a δ function at l0x2/3 ≈ 0.8165l0 ≈ 0.57, and the third peak at twice this value. However, only very few A-mers ever occupy the third layer, even though the data shown in Figure 11 are for NA ) 4: (22) DeGennes, P. G. Scaling Principles in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (23) Daoud, M.; Cotton, J. P. J. Phys. (Paris) 1982, 43, 531-538.

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Figure 9. Radial distribution function GAA(r) of a A-mers and G(r) of micelles plotted vs r, for chain length N ) 8, density φ ) 0.094, and four choices of EAA: EAA ) 3.5 (a), 3.8 (b), 4.0 (c), and 4.2 (d). Corresponding typical snapshot pictures showing the locations of the A-mers of the adsorbed micelles projected onto the xy-plane as shown in the inserts.

the dominant orientation of A-A bonds along the chains hence is parallel rather than perpendicular to the adsorbing substrate surface. The density of the B-mers near the wall, on the other hand, is reduced, as expected: this happens because of excluded volume forces with the wall and with the micellar cores. A distinct peak occurs already at z ≈ 0.7; however, this is due to the B-mers that are “chemical neighbors” of A-mers bound to the substrate. Also, a peak near z ≈ 1.4 due to the (chemical) next-nearest neighbors can be distinguished (Figure 11c). In the outer regime the density profile nB(z) resembles the density profile of polymer brushes.24 It is also evident from the very small values of nA(z), nB(z) for large z that under the conditions studied here almost all chains are bound to the wall, and the concentration of chains that are not adsorbed is vanishing. This point is studied further in Figure 12, where we measure the percentage of A-mers that have z-coordinates less than zmax ) 1: These A-mers are in the range of the wall potential (eq 5) and hence we call them “adsorbed”. It is seen that for the short chains the behavior of the fraction Fads of adsorbed A-mers is nonmonotonic: it first increases with φ, as micelles are formed, but for (24) Grest, G. S.; Murat, M. in ref 9, pp 476-578.

φ > 0.15 it decreases again. The strong decrease of Fads with φ > 0.15 is due to fact that some monomers of the A-part of an adsorbed chain are no longer within the range of the adsorption potential and are thus not counted as adsorbed, in the sense of the definition given above, although they do belong to chains that are attached to the wall. For the longer chains we find a decrease of Fads for large φ. This observation simply tells that at higher densities configurational entropy favors having more and more chains in the solution, rather than crowding the substrate too much with large micelles. Surprisingly, also at fixed φ a similar nonmonotonic variation of Fads with EAA seems to occur; once stable large micelles have formed, the remaining single chains seem to desorb more easily. 5. Dynamic Behavior of Surface Micelles First, we consider the mean square displacement gR(t) of monomers of type R, R ) (A,B):

gR(t) ) 〈[r bRi (t) - b r Ri (0)]2〉

(8)

r Ri (t) labels the lateral position (x,y-coordinates where b only) of the ith effective monomer of type R at time t in

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Figure 10. (a) Radial distribution functions GAA(r), GBB(r), and GAB(r) plotted vs r for the case N ) 8, φ ) 0.094, and EAA ) 4.0. (b) Radial distribution functions GAA(r) plotted vs r for the case N ) 8, φ ) 0.094, and various choices of EAA, as indicated in the figure. (c) Same as (b) but for GBB(r).

Figure 11. (a) Density profile nR(z) plotted vs z, for R ) (A,B), N ) 16, EAA ) 3, and total density φ ) 0.1875 (a box with L ) 32 and D ) 16 is used). (b) Density profile nA(z) plotted vs z, for N ) 16, EAA ) 3, and three choices of φ as indicated in the figure. (c) Same as (b) for nB(z).

the system. The average 〈...〉 in eq 8 could be a time average both over the origin of timesnote that, in thermal

equilibrium, time correlations depend on the relative distance along the time axis only but not on the origin of

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Figure 12. Percentage Fads of adsorbed A-mers plotted vs the total monomer density in the system for N ) 8, NA ) 2, and EAA ) 4 (dots) and for N ) 16, NA ) 4, and EAA ) 3 (triangles).

Figure 13. Histogram of mean square displacements of A-mers and of B-mers after a time of 16 384 Monte Carlo steps per effective monomer (MCS). Data refer to a total density of φ ) 0.25.

timesand over all monomers of type R. Thus, gR(t) can also be written as NR

gRi (t) ∑ i)1

gR(t) ) NR -1

(9)

where giR(t) is the mean square distance travelled in time t by monomer i. Choosing one particular origin of time, we can observe the distribution H RD(r) of distances r )

xgiR(t)

travelled in the time interval t by all the NR monomers. This distribution function is shown in Figure 13 for two choices of EAA, one just below (EAA ) 3.4) and one above (EAA ) 4.0) the threshold for micelle formation. It is seen that H AD(rf0) f 0 for EAA ) 3.4 (i.e., all monomers have moved away from the positions they had taken at time t ) 0). Also, there is little difference between H AD(r) and H BD(r) in this case, A monomers and B monomers being roughly equally mobile. This picture radically has changed after micelles have formed: now, H AD(r) has a δ-function-like peak at the originsmonomers are tightly bound at their positions in the crystalline cores

Figure 14. log-log plot of various mean square displacements vs time for the case N ) 8, EAA ) 4, and total density of φ ) 0.25. Data are shown for single chains (s ) 1) and chains in micelles of size s ) 3 and s ) 15, respectively.

Figure 15. Correlation C(t) of average micelle size with time t for φ ) 0.125, N ) 8, and various choices of EAA.

and can move only very little (like phonon vibrations in a crystal). The tail of H AD(r) simply is due to the few free chains that are not part of the micelles, of course. Note that also the distribution H BD(r) is now much narrowers although this distribution is still peaked at a distance larger than unity, it is remarkable that the tail at large distances (r g 5) has practically been wiped out. The B monomers that belong to the corona region of micelles cannot diffuse very far away from the micellar cores, which are tightly bound to the adsorbing wall; hence, there is no real diffusion for most of the B-mers as well. Figure 14 presents now the mean square displacement averaged over all chains, but distinguishing whether at time t ) 0 a monomer did belong to a single chain (s ) 1) or a micelle of a particular size s. A further distinction is made between the mean square displacement of the inner monomers of a chain (denoted as rCM2) or of its A-part (denoted as rA-CM2). For s ) 1 the center of mass of the A-part (remember that there are NA ) 2 A monomers per chain only) behaves similar to an end monomer of a chain. Hence, it has a

Monte Carlo Study of Asymmetric Block Copolymers

Langmuir, Vol. 15, No. 9, 1999 3241

larger mean square displacement than the inner monomers have, over the time interval studied (Figure 14). For all three mean square displacements, we have a behavior gR(t) ∝ t for s ) 1 at late times, indicating standard diffusive behavior. For s ) 3 and s ) 15, however, we see that all mean square displacements bend over at late times, indicating that there is no free diffusion any longer. Interestingly, for short times the mean square displacement of inner monomers (which is a B monomer irrespective of s, of course) does depend hardly on s. Only at late times does it make a difference whether a chain is bound in a micelle or not, for the monomers that form the corona. For the A monomers, on the other hand, the mobility is dramatically reduced, if they are part of a large micelle. Finally, we consider the correlation function of micelle size, defined as

C(t) )

〈s(t′)s(t′ + t)〉 - 〈s(t′)〉2 〈s2(t′)〉 - 〈s(t′)〉2

(10)

where s(t′) is the average size of micelles observed at time t′ in the simulation box. It is seen (Figure 15) that for EAA e 3.7 this function decays to zero, on the available time scale of a million Monte Carlo steps, while for EAA g 3.8 the size distribution of the micelles is not in full thermal equilibriumslarge micelles that have formed are essentially metastable, frozen-in objects. This means that some of the observations of static properties reported here do not reflect full thermal equilibrium either. However, the same problem presumably concerns the experiments. 6. Concluding Remarks In this paper findings from a Monte Carlo simulation of the formation of surface micelles from short block copolymers at an adsorbing smooth substrate have been described. The model used is a coarse-grained one, lacking any chemical detail, but it is hoped that the generic features of such surface micelles are nevertheless captured. It was found that the general features of the micelles have many features in common with similar models of micelles in bulk solution.10-12 One distinguishing feature, however, is the formation of (mostly) two crystalline layers

in the micellar core, packed adjacent to the wall on top of each other in a close-packed arrangement. This dense structure affects the mobility of the micelles in a dramatic way, which then are essentially frozen-in at the random positions where they are formed. Hence, the model does not yield the nearly regular patterns of the center of masses of the micelles that have been observed sometimes experimentally.8 However, observation of micelle formation with crystalline core in bulk solution has been reported, too,20,21 and we expect that our model predicts properties of micelle formation in such systems in the presence of a strongly adsorbing wall. Related behavior of micelles may have been observed for micelles at the air-water interface.25-27 Other systems to which our work might be useful are ionic surfactants on graphite and gold substrates.28,29 Of course, the present work can be considered as a first step onlysparameters such as the chain length N and the length NA of the associating part need be varied more systematically. Also, the effect of the various parameters characterizing the potentials between the various monomers as well as the wall potential needs study. In reality, roughness of the adsorbing surface (or atomistic corrugation) and chain stiffness may also play an important role. Therefore, it would be too early to try to correlate the present simulations with any experiments in a more specific way. Nevertheless, we hope that the present work will stimulate complementary experimental work as well as analytical theoretical approaches to better understand this self-organization of surfactants at interfaces. Acknowledgment. This work was supported by the Deutsche Forschungsgemeinschaft (DFG), Grant no. 436BUL113/92. We acknowledge stimulating discussions with D. Viduna. LA981387A (25) Zhu, J.; Lennox, R. B.; Eisenberg, A. J. Phys. Chem. 1992, 96, 4727-4730. (26) Li, S.; Clarke, C. J.; Lennox, R. B.; Eisenberg, A. Colloids Surf. 1998, 133, 191-203. (27) Kato, T.; Kameyama, M.; Ehara, M.; Iimura, K. Langmuir 1998, 14, 1786-1798. (28) Manne, S.; Cleveland, J. P.; Gaub, H. E.; Stucky, G. D.; Hansma, P. K. Langmuir 1994, 10, 4409-4413. (29) Jaschke, M.; Butt, H.-J.; Gaub, H. E.; Manne, S. Langmuir 1997, 13, 1381-1384.