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Sep 22, 2014 - Here we report on the behavior of solutions of amphiphilic multiblock copolymers with a large number of blocks and a low fraction of so...
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Multiblock Copolymer Solutions in Contact with a Surface: SelfAssembly, Adsorption, and Percolation Virginie Hugouvieux*,† and Max Kolb‡ †

INRA, UMR1083 SPO, F-34060 Montpellier, France Laboratoire de Physique et Centre Blaise Pascal, CNRS UMR5672, Ecole Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon, France

‡

S Supporting Information *

ABSTRACT: Amphiphilic copolymers are often used as compatibilizing or stabilizing agents, either in solution or at surfaces. In the special case of multiblock copolymers the connectivity of the blocks combines with the antagonistic behavior of the different types of blocks. Here we report on the behavior of solutions of amphiphilic multiblock copolymers with a large number of blocks and a low fraction of solvophobic monomers in contact with an attractive surface. Using lattice Monte Carlo simulations, the influence on the structures of the solvent quality and the type of surface from noninteracting to strongly attractive to the solvophobic monomers can be assessed. In the presence of a surface bulk micelles are formed that are not different in size and shape from the micelles observed in the absence of a surface. When increasing the surface attraction, solvophobic monomers tend to adsorb either as isolated blocks or forming surface micelles. Evidence is given of a surface concentration threshold above which surface micelles can form due to microphase separation. These surface micelles are in equilibrium with bulk micelles, some of which are connected to the surface through a path of either hydrophobic and/or hydrophilic blocks or hydrophobic cross-links, or both. The size distributions of bulk and connected micelles are similar. With increasing surface concentration surface micelles get organized due to the steric repulsion between core−shell surface micelles. Moreover, these organized surface micelles percolate. The connected micelles form a concentrated layer parallel to the attractive surface. In addition, these systems are governed by two very different time scales: The fast one leads to micellar self-assembly in the bulk and at the surface while the slow one prevents the system from reaching equilibrium in the course of the simulations and corresponds to the transfer of copolymers from the bulk to the attractive surface.

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INTRODUCTION Block copolymers consisting of solvophobic and solvophilic blocks show a versatile behavior. This versatility is due to the various types of constitutive monomers (hydrophobic, hydrophilic, charged), to the length and number of blocks (from diblock to multiblock and random), and to their architecture which may range from linear to branched and starlike. The variety of available macromolecules gives rise to a number of applications,1 which rely on the ability of amphiphilic block copolymers to self-assemble in solution and to adsorb at surfaces or interfaces. Applications of these properties range from drug encapsulation and compatibilization, wetting agents, and rheology modifiers to emulsion, foam, and nanoparticle stabilizers.2−4 The special case of alternating multiblock copolymers consisting of two types of monomers is particularly interesting for two main reasons. On one hand, multiblock copolymers can be seen as a simple model of some blocky polysaccharides5 or proteins.6 On the other hand, the connectivity of multiblock copolymers gives rise to self-assembly and gelation properties © 2014 American Chemical Society

which are not available from di- and triblock copolymers. The self-assembly of multiblock copolymers in dilute solution was studied using scaling arguments7 and simulations8,9 and showed the formation of pearl necklaces of micelles7−9 as well as tubular and layered structures,9 depending on the ratio of hydrophobic to hydrophilic monomers in the chains and the quality of the solvent toward hydrophobic monomers. Regarding semidilute solutions, where both intra- and intermolecular interactions are present, multiblock copolymers perform microphase separation and form a gel network as demonstrated using a two-dimensional model calculation.10 Gindy et al.11 showed that symmetric multiblock copolymers may either perform microphase separation or macroscopic phase separation depending on the ratio of the number and length of the blocks. From the experimental side microphase separation of graft copolymers was studied as a function of the Received: July 24, 2014 Revised: September 19, 2014 Published: September 22, 2014 12400

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hydrophilic and hydrophobic block lengths.12 More recently, the formation of intermolecular micelles and percolating networks was demonstrated using Monte Carlo simulations,13 and the structure of the percolating network was shown to depend strongly on the ratio of hydrophobic monomers. Regarding the properties of multiblock copolymers at a surface or an interface, several studies were performed using mean-field theory, scaling arguments, and simulations. The compatibilizing effect of block copolymers at a surface or a penetrable interface was studied: The effect on the properties of the adsorbed layer of the random or blocky pattern, the number of blocks (di-, tri-, and multiblock), their lengths, and the position of the stickers (end position or not) in the case of triblock copolymers was investigated by means of Monte Carlo simulations.14−16 Meanfield arguments were used to determine the adsorption threshold of symmetric multiblock copolymers as a function of the number and length of the blocks.17 More recently, the effects of the block length and segregation regime on the size of the chains perpendicular and parallel to the surface were studied using a combination of scaling arguments and simulations.18 The kinetics of reorganization of the copolymers perpendicular and parallel to the surface were also investigated, showing that rearrangements are faster perpendicular to the surface.19 Similar methods were used to study the effect of the block and chain lengths on the critical adsorption energy and the fraction of adsorbed monomers.20 The influence of the surface concentration regime (from dilute to saturated) and of the solvent selectivity was shown to affect the properties of the adsorbed layer, with conformations of the copolymers ranging from large loops to hairpins and eventually formation of micelles in the bulk.21,22 Following these results, the phase diagram and surface pressure of the different surface regimes were derived using scaling arguments and compared to the properties of adsorbed layers of proteins.6 Although the behavior of multiblock copolymers in solution or at a surface was investigated, few studies were dedicated to the case of a multiblock copolymer solution in contact with a surface, accounting for the coexistence of copolymers in solution and at the surface. In the case of diblock copolymers, Zhan et al.23 and more recently Cavallo et al.24 dealt with the interplay between micelle formation in solution and adsorption. The case of multiblock copolymer solutions in contact with a surface was previously investigated using theoretical calculations either in solutions below the cmc25,26 or in melts,27 with a focus on the composition profiles of and interaction curves between layers of adsorbed multiblock copolymers. These studies however did not explore the possible formation of micelles. In the present work we focus on the competition between self-assembly in solution and surface adsorption of multiblock copolymers. With the restriction that actual proteins seldom consist of large blocks of hydrophilic or hydrophobic monomers, this situation can be seen as a simple model of protein-stabilized foams and emulsions,28 where proteins in solution and partially denatured proteins adsorbed at the interface coexist. It is also important to understand the interplay between micelle formation in solution and adsorption of copolymers in view of tuning surface properties after adsorption. In the case of multiblock copolymer solutions near a surface, the behavior of the system is governed by several driving forces. In solution the monomer−solvent and monomer−monomer interactions govern the self-assembly of the copolymers into micelles whenever the bulk concentration is higher than the critical micelle concentration. Near the

surface or the interface the monomer−surface interactions lead to the adsorption of the copolymers when the surface attraction to one of the blocks is strong enough. Because of the interplay between the monomer−solvent, monomer−monomer, and/or monomer−surface interactions and the connectivity between hydrophobic or hydrophilic blocks, multiblock copolymers may adsorb and/or self-assemble spontaneously in a variety of structures, such as spherical, tubular, or lamellar core−shell micelles, either in solution or at the surface. In order to improve our understanding of the interplay between self-assembly, either in solution or at the surface, and adsorption, we perform Monte Carlo simulations of the behavior of a solution of linear multiblock copolymers made of a large number of regular alternating hydrophobic (H) and hydrophilic (P) blocks in contact with a surface which may be either noninteracting (repulsive) or hydrophobic (i.e., attractive for the H monomers). We focus on the self-assembled structures observed in solution and at the surface and on the threshold for the formation of the adsorbed structures. The questions of the two-dimensional organization of the adsorbed selfassembled structures and the distribution of the copolymers perpendicular to the surface are assessed. We will show that the different processes (self-assembly, adsorption, distribution of the copolymers in the system) involve different time scales and that the system under study has not reached equilibrium with regards to some properties. The paper is organized as follows: The first section is dedicated to the model, the simulation method, and the definition of the properties computed for characterizing the behavior of the system. Then the results regarding the selfassembly, adsorption, and percolation of the copolymers are presented, discussed, and compared with previous studies.

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SIMULATION MODEL AND METHOD The model was previously described in the case of dilute and semidilute solutions of multiblock copolymers.9,13 We consider solutions of linear, regularly alternating multiblock copolymers made of two kinds of monomers: hydrophobic (H) or hydrophilic/polar (P). In the present work these solutions are in contact with a noninteracting or hydrophobic surface. The pattern of the blocks is denoted as (HBHPBP)n, where n is the number of pattern repeats and BH (BP) is the length of the H (P) blocks. The degree of polymerization of the chain is Nm = n × (BH + BP), and the hydrophobic substitution rate is defined as Psub = BH/(BH + BP). We focus on the case of long chains with a large number of blocks, with n ranging from 5 (Nm = 100) to 15 (Nm = 600). For the simulations, we use an efficient lattice Monte Carlo (MC) model, which we introduced in a previous article.9 In this model polymer chains are discretized on a face-centered cubic lattice. This allows for a large number of nearest neighbors and bond angles. A lattice site may be occupied by zero, one, or two monomers. Hence, a lattice site represents the volume of two monomers. Adjacent monomers in a polymer occupy either the same site on the lattice or nearest-neighbor sites. Note that double occupancy of a lattice site is restricted to adjacent monomers along a polymer.9 All monomers strictly obey to the excluded-volume constraint (which means in the present case that a lattice site cannot be occupied by more than two monomers). Moreover, the polymers are never allowed to cross each other. 12401

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surface interaction energy εs ranged from 0 to −0.55. Hence, the system was studied both below and above the critical micelle concentration,13 and the surface properties ranged from noninteracting to strongly attractive to the H monomers. In the present study we focused on a single substitution rate, Psub = 0.2 (with BH = 4 and BP = 16), which corresponds to a good screening of the hydrophobic cores by the hydrophilic coronae13 in the case of fully flexible copolymers. More information about the simulation details and the parameters of interest is given in the Supporting Information. Properties of the Simulated Systems. All static properties can be computed from the set of independent equilibrium configurations, which give a quantitative assessment of the structural properties. A cluster is defined as a set of interacting copolymers. Two copolymers interact whenever at least one H monomer of the first copolymer is a nearest neighbor to an H monomer of the second copolymer. A cluster comprises all the monomers (H and P) of the involved chains. A hydrophobic core is defined as a set of interacting H monomers, ignoring the P monomers and the connectivity of the chains. Note that a cluster may contain several hydrophobic cores. If we consider a single chain, it always belongs to a single cluster for connectivity reasons, but it may contain as many hydrophobic cores as the number of H blocks in a chain, especially in good solvent conditions. NcH is the number of H monomers in a given core, and W(NcH) is the fraction of H monomers in cores of size NcH. A copolymer (or a core or a cluster) can be classified in terms of three different categories depending on its position toward the attractive surface. A copolymer/core/cluster is adsorbed if at least one of its H monomers interacts with the attractive surface (i.e., at least one of the H monomers of the copolymer/core/ cluster is a nearest neighbor to a site of the attractive surface). The second option for a copolymer/core is to be connected to the surface: in this case the copolymer/core is not adsorbed but belongs to an adsorbed cluster. The last case corresponds to bulk structures: bulk copolymers/cores do not belong to an adsorbed cluster; bulk clusters are not adsorbed. With these definitions of adsorbed, connected and bulk entities we can classify the contributions from (i) the copolymers that are directly adsorbed to the surface, (ii) the adsorbed layer (including adsorbed and nonadsorbed copolymers belonging to the same adsorbed cluster), and (iii) the nonadsorbed copolymers/clusters. A cluster is considered to be percolating if it spans the whole simulation box, i.e., if one or more of its dimensions are larger than the size of the box. A configuration percolates if it contains at least one percolating cluster. For a given set of conditions the probability of percolation, Pperc, is defined as the fraction of percolating configurations. We take Pperc = 0.5 to be the onset of gelation.

The solvent is described implicitly through the hydrophobic interaction. The latter is mimicked by an effective nearestneighbor attractive interaction between the H monomers, which leads to the expulsion of the solvent from the vicinity of the hydrophobic monomers. This attractive force acts between any two H monomers that occupy the same or two nearestneighbor sites (called interacting monomers in the following) and is given by the interaction energy εi < 0, expressed in units of the thermal energy kBT. Increasing the absolute value of εi corresponds to a strengthening of the H−H interaction and hence is experimentally equivalent to a lowering of the temperature (except for the case of systems with a lower critical solution temperature (LCST), for which it corresponds to an increase in temperature29). The P monomers interact neither with the P nor with the H monomers. The copolymer solution is confined between two impenetrable and parallel surfaces, one of which can be tuned from noninteracting to attractive to the H monomers and the other acting as a repulsive wall. The size of the simulation box is Lc in the x and y directions and Lz = (2/3)1/2Lc in the z direction, which corresponds to the distance between the walls. Periodic boundary conditions are restricted to the directions parallel to the surface. The surface attraction is restricted to the H monomers that are nearest neighbors to the attractive surface, and its strength is denoted as εs (expressed in kBT units). In the following a copolymer is said to be adsorbed if at least one of its H monomers is nearest-neighbor with an attractive surface site and hence interacts with the attractive surface. Note that (i) εs = εi means that a H monomer is equally attracted to a nearestneighbor attractive surface site and to a nearest-neighbor nonsurface lattice site occupied by two H monomers and (ii) due to the FCC lattice, whenever a H monomer is a nearest neighbor of the attractive surface it actually interacts with three attractive surface lattice sites. Dynamics is implemented by randomly selecting monomers and attempting to move them to nearest-neighbor lattice sites. Note that the present cell polymer dynamics model only uses local moves, which were carefully chosen in order to produce the correct kinetics, at least on scales large compared to the lattice spacing. The details of the allowed moves can be found in a previous work9 where we checked that the model correctly reproduces the time correlation functions of the exactly known eigenmodes of the Rouse model.30 Hence, the present Monte Carlo simulations do not use fictitious dynamics, and kinetic and out of equilibrium data can be obtained. Moves are accepted if the new configuration again respects bond length and excluded volume constraints. For the interacting hydrophobic monomers acceptance of the trial moves is also ruled by the Metropolis criterion.9 Simulations were performed for different values of copolymer chain length, volume fraction, solvent quality, and surface attraction. The volume fraction φ is defined as the ratio of the volume occupied by the monomers to the total volume available in the simulation box. Since each lattice site can be occupied by up to two monomers, the total available volume is twice the number of lattice sites. We studied monodisperse systems with volume fractions of 0.01 or 0.02 and containing Np = 100 or 200 copolymers with chain lengths Nm = 100, 300, or 600. In the simulated systems, the distance Lz = (2/3)1/2Lc between the walls ranged from 57 to 74, which is at least 4 times larger than Rg for Nm = 600 and εi = 0 at the studied volume fractions. The H monomer−H monomer interaction energy was varied between εi = −0.51 and −0.55 while the

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RESULTS It has been shown for isolated copolymers that the substitution rate determines the pattern developing, through a balance between energetic (hydrophobic) and entropic (hydrophilic) contributions.9 In more concentrated solutions (for φ between 0.5% and 25%), we gave evidence for the formation of different types of micelles and gels depending on the substitution rate of the copolymers. The present study focuses on the behavior of a solution of copolymers in contact with a surface, either 12402

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Figure 1. Snapshots of the system for Np = 200, Nm = 600, εi = εs = −0.55, a total volume fraction φ = 0.02, and after 1.6 × 108 MCS. Top: side view perpendicular to the attractive surface; the black line at the top shows the position of the repulsive wall, and periodic boundary conditions apply in the directions parallel to the surfaces. Bottom: view parallel to the attractive surface (only the three layers closest to the surface are shown). H monomers are shown as red beads while P monomers are shown as gray beads. Note that each H or P bead occupies the same volume in the simulations. In the snapshots they are depicted with different sizes for clarity.

noninteracting or attractive to the H monomers, in the case of a low substitution rate Psub = 0.2 with BH = 4 and BP = 16. Figure 1 shows two snapshots of a typical configuration of a system above the bulk critical micelle concentration (for the concentration shown, the cmc occurs between εi = −0.51 and −0.52, as determined previously13) and in the presence of a strongly attractive surface. The top snapshot shows the whole system perpendicularly to the surface (side view, the attractive surface is at the bottom, the noninteracting one at the top) while the bottom snapshot is taken parallel to the attractive surface and only shows the three layers neighboring the attractive surface. In the side view and going from the bottom to the top of the picture we observe that the 2−3 layers at the surface are mainly populated by H monomers while the next 4−6 layers are mainly occupied by P monomers. The following layers are occupied by bulk and connected core−shell micelles together with isolated H blocks. Then the system seems to be depleted, probably due to the length of the copolymers and the presence of the thick adsorbed layer which may act as a rough surface. Farther from the surface, core−shell micelles and isolated H blocks are again present and are similar to what was observed in semidilute solution of the same copolymers.13 The layers close to the upper surface are again depleted. Most copolymers seem to be connected to the surface (through covalent bonds or interacting H monomers), even though they are not directly adsorbed: They may be connected either through P blocks or/and via interacting monomers. When looking at the surface (Figure 1, bottom), aggregates of H monomers are present due to the surface attraction and the monomer−monomer attraction. In the absence of surface attraction (not shown) no copolymer is adsorbed, and depletion occurs in the vicinity of the surface. Away from the surface the copolymers self-assemble into core−shell micelles, and isolated blocks are also present, as previously observed in semidilute solutions.13

In the following we perform a quantitative analysis of the self-assembly, adsorption, and percolation properties of the system. Self-Assembly. As shown in the snapshots the behavior of the system results from an interplay between adsorption at the surface and self-assembly in the bulk and at the surface. The self-assembly features of the copolymers in solution in the presence of an attractive surface are shown in Figure 2. The distribution W(NcH) of H monomers over different core sizes NcH in the different types of cores (see previous section for the definition of adsorbed, connected, and bulk cores) is shown for a total volume fraction of 2%. We notice that the distributions of H monomers in the bulk and connected cores have the same shape: Isolated blocks are present in equilibrium with H cores

Figure 2. W(NcH) for the different types of hydrophobic cores. The data are for Np = 100, Nm = 600, εi = εs = −0.55, φ = 0.02, and after 3 × 107 MCS. 12403

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monomers in the bulk, which is balanced by the entropic penalty for accommodating the P blocks around the H cores. This is in agreement with what was previously shown in the case of bulk cores.13 Connected cores differ from bulk cores by the fact that they are connected (but not adsorbed) to the surface through a path of covalent and/or interacting monomers. Hence, connected cores are expected to be present in the system whenever bulk cores are present. The presence of adsorbed cores for εi = −0.51 can be attributed to a higher effective concentration near the surface when the latter is attractive enough as will be shown later (see Figure 6). The time evolution of the self-assembled hydrophobic cores is shown in Figure 4. We notice that the overall shape of the

of a well-defined size, as shown by the peak around 45 H monomers. Adsorbed cores are also present with a typical size of 20 H monomers. We also checked that in the present simulations the presence of a surface, either attractive or noninteracting, does not affect the bulk self-assembly of multiblock copolymers into core−shell micelles. This is shown in Figure S1 of the Supporting Information. The influence of chain length on the size distribution of bulk and surface micelles is also assessed (see Figure S2 of the Supporting Information). The decreasing typical size of bulk micelles with increasing chain length can be rationalized by the entropic penalty related to internal and end P blocks (see Supporting Information). The influence of varying the bulk and surface interactions below or above the critical micelle concentration (cmc)13 is shown in Figure 3. This figure presents the distribution of H

Figure 4. Time evolution of the W(NcH) distributions of adsorbed (black), connected (red), and bulk (blue) hydrophobic cores. The curves are shown for Np = 200, Nm = 600, φ ≈ 2%, and εi = εs = −0.55. The different line types correspond to different simulation times: 2 × 107 MCS (solid), 4 × 107 MCS (dashed), 6 × 107 MCS (dash-dotted), 8 × 107 MCS (dotted), and 1.7 × 108 MCS (double dot-dashed).

Figure 3. Distribution of H monomers over cores of size NcH below (εi = −0.51, left) and above (εi = −0.55, right) the energy threshold for micelle formation (which is between εi = −0.51 and −0.52) determined in the bulk13 and for the three types of hydrophobic cores: adsorbed (black circles), connected (red squares), and bulk (blue triangles) cores. The distributions are shown for Np = 100 and Nm = 600 and for two values of the surface attraction: εs = −0.51 (open symbols) and εs = −0.55 (full symbols). The total volume fraction is 2%.

distributions of connected and bulk cores does not evolve much with time: They show a well-defined peak corresponding to the micelles and a high intensity for small aggregation numbers which is due to the presence of isolated blocks in equilibrium with the core−shell micelles. The adsorbed cores are also present whatever the simulation time; however, their average size shifts to slightly higher values between 2 × 107 and 4 × 107 MCS. Hence, the self-assembled structures are preserved for the different simulation times. Yet a variation of the relative contributions of connected and bulk cores is observed, with an increase of the proportion of bulk cores and a decrease of the proportion of connected cores between 2 × 107 and 4 × 107 MCS. After 6 × 107 MCS the amounts of adsorbed, connected, and bulk cores evolve very slowly with a weak logarithmic time dependence, which can be seen by integrating the W(NcH) distributions for different simulation times. The time evolution of the distributions of H monomers over cores of sizes NcH is similar for a smaller system (Np = 100 and same volume fraction). We also notice that the relative proportions of bulk and connected cores evolve with copolymer length Nm and total volume fraction (data not shown). For short chains (Nm = 100) connected cores are absent or much less present than bulk cores (by a factor decreasing from 100 to 10 when increasing concentration from 1 to 2%). For Nm = 300 (600) the ratio of bulk to connected cores is of the order of 50 (10) at φ = 0.01 and of the order of 0.5 (0.3) at φ = 0.02. Hence, the proportion of connected cores increases with increasing chain length and

monomers over cores of different sizes for copolymers with Nm = 600, for two values of the monomer−monomer interaction (εi = −0.51, left, and εi = −0.55, right) and two values of the monomer−surface interaction (εs = −0.51, empty symbols, and εs = −0.55, full symbols). Note that for a concentration of 2% at εi = −0.51 the simulated system is below the cmc, while it is above the cmc at εi = −0.55.13 For the lower value of the monomer−monomer interaction no bulk or connected cores are observed: The corresponding W(NcH) distributions show a steep and monotonous decrease. However, the formation of adsorbed cores appears as an incipient shoulder in the distribution around 20 H monomers, which becomes more pronounced when increasing the monomer−surface interaction. On the other hand, for εi = −0.55 the W(NHc) distributions of adsorbed, connected, and bulk cores show a well-defined peak, around 20 H monomers for the adsorbed cores and between 45 and 50 H monomers for the connected and bulk cores. The size distributions of both bulk and connected cores have similar shapes, with the coexistence of isolated blocks (NcH ≤ 8) and H cores with a well-defined size (NcH ≈ 45). The presence of bulk and connected cores is directly related to the strength of the interaction between H 12404

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than −0.51 show the existence of two classes of cluster sizes: The first one consists of small clusters (⟨Rg⟩ ≈ 14 (10) at εi = −0.51 (−0.55) and the second one of much larger clusters (⟨Rg⟩ ≈ 61 (59) at εi = −0.51 (−0.55). The average sizes of the clusters increase with increasing interaction strength, from ⟨Rg⟩ ≈ 21 at εi = −0.51 to ⟨Rg⟩ ≈ 27 at εi = −0.55. This shows that the class of large clusters becomes relatively more abundant when increasing the interaction. Also, at εi = −0.55 the average radii of gyration of both classes of clusters decrease with simulation time, in agreement with the fact that the depleted layer increases slightly at εi = −0.51 and noticeably at εi = −0.55. Hence, the clusters contract with time and the depleted layer near the surfaces becomes thicker. The evolution of this layer probably results from the entropic effects due to the presence of the surface (already present at εi = 0) and from the energetic effects which lead to the association and contraction of the clusters of copolymers. Changing the surface from noninteracting to attractive has important consequences on the behavior of the system. As expected, H monomers tend to adsorb as the attraction of the surface becomes strong enough. This is shown in Figure 6, where the lattice sites at z = Lz/2 = 37.5 are attractive to the H monomers (the lattice sites at z = Lz/2 are blocked: they cannot be occupied by any monomer) whenever εs is strictly negative (the wall at z = −Lz/2 is always noninteracting). The top graph shows the distribution of all monomers along the zdirection perpendicular to the surface for different values of the surface attraction and monomer−monomer interaction. As expected, the density profiles in the case of an attractive surface strongly differs from the ones shown in Figure 5. They show a large excess of monomers in the layers close to the attractive surface. This surface excess increases noticeably when increasing surface attraction from εs = −0.51 to −0.55. On the contrary, increasing the monomer−monomer interaction results only in a slight increase of the surface excess. Hence, the surface excess increases when increasing the surface attraction. On the contrary, increasing the bulk interaction leads to the formation of core−shell structures and modulates the transfer of copolymers from the bulk to the surface: Indeed at εi = −0.51 copolymers in the bulk are not self-assembled while they form core−shell micelles at εi = −0.55. In the latter case the diffusion of the self-assembled copolymers is slower as compared to the free copolymers, as stated by Zhan et al. in the case of diblock copolymers.23 The latter argument is also consistent with the presence of a minimum in ρ(z) around z = 0.6Lz/2 ≈ 20 at εi = −0.55. The bottom graph in Figure 6 shows the separate contributions of the H and P monomers to the z-profile for two values of εi and εs. The H monomers are directly present at the attractive surface as shown by the intense and thin peak close to z = Lz/2 = 37.5 while the maximum of the peak of the P monomers is slightly shifted toward the bulk. This is consistent with previous results on the adsorption profiles of diblock copolymers.23,24 The profile of the H monomers also shows a secondary peak close to the surface: Due to the connectivity of the multiblock copolymers, the H blocks adjacent in the copolymers to the adsorbed H blocks are expected to be in the vicinity of the surface. Note that computing the z-profiles of the adsorbed, connected, and bulk H cores shows that the peak of H monomers at the surface is due to the adsorbed cores, while the secondary peak is totally explained by the presence of connected cores (data not shown). We mentioned earlier that the presence of copolymers at the surface depends on the transfer of monomers from the bulk to

concentration: This can be rationalized by the enhanced connectivity due to longer chains and the reduced distance between copolymers due to concentration. Distribution of the Copolymers in Solution and at the Surface. Another interesting property of the system is the distribution of monomers perpendicular to the surface, which we call z-profile ρ(z). The value of ρ(z) corresponds to the fraction of the volume of the z-layer occupied by monomers. For a noninteracting surface the z profiles give insight into the depletion effect of the walls. This is shown in Figure 5 for

Figure 5. Density profiles of the copolymers in contact with a noninteracting surface, for different values of the monomer−monomer interaction. The results are shown for Np = 200, Nm = 600, φ ≈ 2%, and εs = 0 (noninteracting surface).

different values of εi from good to poor solvent. The behavior is similar for the two system sizes, with Np = 100 (data not shown) and Np = 200 and the same volume fraction. We notice that the profiles are symmetric. For a given value of εi the concentration away from the surfaces is roughly constant apart from a slight concentration excess near the depleted layers: This latter concentration excess tends to disappear as equilibration takes place. From εi = 0 to εi = −0.55 the bulk concentration varies between 3.5% and 4.2% (we mean here the concentration per layer away from the surfaces), increasing as the depletion layer becomes thicker. The monomer concentration near the surfaces is strongly reduced as compared to the bulk density and reaches ρ(z) = 0 at the walls. This corresponds to the well-known depletion effect: Close to the surface the entropy of the copolymers is reduced, and this results in an effective repulsion from the surface, leading to a depletion layer.31 In the simulations the thickness of the depletion layer near the surfaces tends to increase with increasing strength of the monomer−monomer interaction. Below the overlap concentration the thickness of the depletion layer is known to scale with the radius of the objects.32 In our case, the average radius of gyration of the copolymers decreases from around 15 at εi = 0 to 11 at εi = −0.55 (for Nm = 600). Hence, this scaling argument is not valid in our case. However, at εi = −0.55 the copolymers no more behave as individual chains but are part of self-assembled structures which develop with increasing interaction strength. Consequently, the depletion thickness seems to be related to the size of the supramolecular structures, whose radii of gyration are much larger than those of individual chains. The distribution of the radii of gyration of the clusters (data not shown) for εi stronger 12405

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peak which may correspond to a tertiary layer. The minimum between the secondary (and the developing tertiary) layer and the bulk shifts away from the attractive surface as copolymers transfer from the bulk to the secondary and tertiary layers. The distribution of copolymers in the system has not reached equilibrium in the course of the simulations as can be seen from the evolution of the secondary and third layers. Copolymers are still transferring to the surface and the layers connected to the surface, due to the strong surface attraction and the connectivity of the multiblocks which drag the chains toward the surface. The secondary layer is observed for the three chain lengths of interest, and its position does not depend on chain length (see Figure 8). The position of the secondary layer is

Figure 6. Density profiles of the copolymers in the direction perpendicular to the surface for Np = 100, Nm = 600, and a total volume fraction of monomers of 2%, after 3 × 107 MCS. Note that z = 37.5 corresponds to the attractive surface while z = −37.5 corresponds to the noninteracting one. (a) Density profiles of all monomers for two values of εi and εs. (b) Separate contributions of the H (dotted lines) and P (solid lines) monomers to the density profile for the same values of εi and εs. Note that the y-scale is logarithmic in this plot. The color code is the same for both plots.

Figure 8. Comparison of the z-profiles of the H monomers for different values of Nm with a focus on the adsorbed and secondary layers and for εi = εs = −0.55 and φ ≈ 0.02 and after 108 MCS. In this plot the attractive surface is at z = 0.

the surface. The kinetics of this transfer is shown in Figure 7. While the surface excess does not vary much with simulation

preserved for the three chain lengths because this position is mainly influenced by the size of the surface micelles and the length of the P blocks, which are constant. The same phenomenon is observed for εi = −0.51 when surface micelles are present (data not shown). Surface Concentration Threshold for the Formation of Surface Micelles. The evolution of surface concentration with simulation time is shown in Figure 9. We define the surface concentration ρs as the proportion of the surface occupied by H or P monomers. The different curves correspond to different chain lengths (Nm = 100, 300, and 600) and different values of the bulk interaction (εi = −0.51 and −0.55) and the surface interaction (εs = −0.41, −0.51, and −0.55). We notice that the surface concentration first increases quickly and then reaches a plateau. The first increase is longer when the chain length is smaller: When a chain is in contact with the surface, all its H blocks will very soon touch the surface and hence the surface concentration increases faster with long chains. For Nm = 300 and 600 the measured plateau values of ρs are similar for fixed values of εi and εs. Note that for Nm = 100 the plateau value is lower: Reaching a given surface concentration with short chains corresponds to a higher translational entropy penalty than with long chains. Moreover, the plateau value is lower for εi = −0.51 than for εi = −0.55 (other parameters being fixed): In the latter case, the chains are more compact due to the stronger attraction between H monomers; hence, more chains can sit at the surface. The

Figure 7. Time evolution of the z-profile of the system for all monomers (solid lines) and only H monomers (dashed ines). The color code is the same for both distributions. The results are shown for Np = 200, Nm = 600, εi = εs = −0.55, and φ ≈ 0.02.

time other features of the density profiles evolve with time. The secondary layer of micelles (secondary peak near the attractive surface) is more than doubled between 2 × 107 MCS and 108 MCS. The bulk content of copolymers is conversely reduced. Moreover, a shoulder develops on the left side of the secondary 12406

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Figure 9. Surface concentration versus time for different chain lengths (black: Nm = 100; red: Nm = 300; green: Nm = 600), bulk and surface interaction strengths (diamonds: εi = −0.55, εs = −0.41; squares: εi = −0.55, εs = −0.51; circles: εi = −0.55, εs = −0.55; crosses: εi = −0.51, εs = −0.51; triangles: εi = −0.51, εs = −0.55) and concentrations (small open symbols: φ ≈ 0.01; full symbols: φ ≈ 0.02). The large open symbols correspond for each set of parameters to the first appearance of surface micelles with a well-defined size as shown in W(NcH) (see Figure S2 in Supporting Information). It defines ρcs. Figure 10. Pair correlation function of the H monomers adsorbed to the attractive surface (correlations are computed for any pair of adsorbed H monomers, other monomers are not accounted for in this calculation) for different values of εs and after 2 × 107 MCS. The results are shown for Np = 200, Nm = 600, ρ = 0.02, and εi = −0.55. Inset: g(r) normalized by the surface concentration of H monomers ρHs (in the main graph, g(r) tends to ρHs at large distances) for εs ≤ −0.41. The same color code is used in both graphs.

entropic penalty of accommodating many chains at the surface is better compensated by the association energy of H monomers at εi = −0.55. The first appearance of surface micelles is shown for each set of parameters as a large open symbol (except for two cases where no micelles are formed: Nm = 600, εi = −0.55, εs = −0.41 and Nm = 600, εi = −0.51, εs = −0.51) and gives access to the surface concentration threshold for the formation of surface micelles, ρcs. The existence of such a critical concentration was previously demonstrated in the case of diblock copolymers.33 In the present study ρcs depends on the value of εi: For εi = −0.55, ρcs is between 0.257 and 0.268 while for εi = −0.51 we find ρcs = 0.3. When the attraction between the H monomers is weaker, a higher surface concentration is needed for the surface micelles to form. In the simulated range these values of ρcs are independent of the chain lengths and the surface attraction, provided the latter is strong enough. (See for instance the case εi = −0.55 with εs = −0.41 for which ρs = 0.26 is not reached. For εi = εs = −0.51, ρs = 0.30 is not reached neither due to the weak surface attraction.) Regarding the influence of the bulk concentration, it may be a limiting factor whenever the amount of copolymers in the bulk does not enable the surface concentration threshold to be reached upon adsorption. Surface Organization and Percolation. With increasing surface attraction the surface concentration increases as shown in the density profiles. The pair correlation functions of the adsorbed H monomers can give insight into the organization at the surface, by computing the distance between any pair of adsorbed H monomers. The pair correlation functions g(r) are shown in Figure 10 for different values of εs and in Figure 11 for different simulation times at εs = −0.55. Starting from εs = −0.41 and with increasing εs, a correlation hole followed by a correlation peak at higher distances (r ≈ 7.5) appear and grow. The value of g(r) at large distances corresponds to the average surface concentration in H monomers and shows that the surface concentration increases as the surface attraction becomes stronger, in agreement with Figure 6a. The presence of correlations in g(r) shows that the surface cores tend to get

Figure 11. Pair correlation function of the H monomers adsorbed to the attractive surface at εs = −0.55 and for different simulation times. The results are shown for Np = 200, Nm = 600, φ = 0.02, and εi = −0.55. The black line with circles is for Np = 100, Nm = 600, φ = 0.02, εi = −0.51, εs = −0.55, and after 4 × 107 MCS.

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presence of a surface (either interacting or not) does not affect the formation and equilibrium size of bulk micelles. Whenever the surface is attractive, H monomers tend to adsorb to the surface either as isolated blocks or as surface micelles with a rather well-defined size. The size of these surface micelles is approximately half the size of the bulk micelles. The presence of surface micelles is observed whenever bulk micelles are present; however, these surface micelles may also be present slightly below the bulk critical micelle concentration as shown in Figure 3. The existence of surface micelles below the bulk cmc was previously demonstrated in the case of diblock copolymers using scaling arguments34 and Monte Carlo simulations.24 We notice that for fixed values of εi the presence of surface micelles is observed only if the surface concentration (the value of ρ(z) for the layer directly interacting with the attractive surface) exceeds a threshold value ρcs : For instance, at εi = −0.55, ρcs ≈ 0.26 for Nm = 100, 300, and 600 and for εs = −0.55 or −0.51; at εi = −0.51 below the bulk cmc the threshold is obtained for a higher value ρcs ≈ 0.30. These values are obtained by following the time evolution of both the W(NcH) distribution of adsorbed cores and the concentration of the adsorbed layer (see Figure 9). The threshold surface concentration is defined as the lowest surface concentration for which surface micelles are present. From this we can deduce that the formation of surface micelles proceeds in a similar way as the formation of bulk micelles with a critical surface concentration below which micelles are absent and above which microphase separation occurs. The existence of a critical concentration for the formation of surface micelles was also demonstrated using simulations of asymmetric diblock copolymers.33 Moreover, the existence of a critical surface micelle concentration different from the bulk critical micelle concentration was predicted by Ligoure34 using scaling arguments, and the order of the two critical concentrations was shown to depend on the contact angle of the adsorbing blocks. The excess concentration at the surface also leads the surface micelles to get organized as shown in the corresponding pair correlation function (see Figures 10 and 11). As the surface concentration increases, a correlation peak develops which is a signature of a preferred distance between the adsorbed H cores. Note that the correlation hole is already present for low surface interactions: This is probably due to the presence of the P blocks bound to the adsorbed H blocks which prevent other H blocks from adsorbing near the already adsorbed H blocks. A similar behavior was observed experimentally in the case of surfactant micelles at the air−water interface:35 Using grazingincidence small-angle X-ray diffraction evidence was given that surface micelles organize following a hexagonal pattern. Also using microscopy and simulations, Spatz et al.36 were able to demonstrate that surface micelles of diblock copolymers get ordered under certain conditions. Surface micelles were also observed in the case of nonionic diblock copolymers37 which form a two-dimensional lattice with a high degree of ordering. With increasing polymer concentration the authors also observed a transition from circular to ribbon-like surface micelles. While we were able to observe the ordering of the surface micelles, no structural transition was observed yet in the simulations. In solution we previously gave evidence of the formation of ribbon-like structures for Psub = 0.5,13 where the balance between energetic and entropic contributions is very different from Psub = 0.2. Hence, increasing the ratio of

organized when the surface concentration (and attraction) increases. The presence of a correlation peak reveals the existence of a well-defined distance between the surface cores while the correlation hole denotes the absence of H monomers inbetween the surface cores. The correlation peak at εs = −0.41 is broader and occurs at lower distances as compared to its width and position at εs = −0.51 or −0.55 (see inset in Figure 10): This is due to the presence of surface cores in the latter case, which are absent at εs = −0.41. These surface cores are made of H monomers surrounded by P blocks. The presence of P blocks bound to adsorbed H blocks leads to steric constraints. Hence, the surface micelles get organized in order to optimize the balance between the energy of the self-assembled adsorbed H cores and the entropy of the surrounding covalently bound P blocks, which are constrained by their connectivity to the surface cores and by the presence of the surface. Regarding the time evolution of g(r) at εs = −0.55, we notice that the position and width of the correlation peak are not affected. This is in agreement with the preserved size distribution of the adsorbed cores with simulation time. However, the average surface concentration increases and becomes stable after 7 × 107 MCS. A secondary correlation peak tends to develop around r = 15 at εs = −0.55 and for long simulation times. It reveals that the system gets organized up to a length scale corresponding to the second-nearest-neighbor cores. The presence of a correlation hole and a correlation peak is also observed for εi = −0.51 (black line with symbols in Figure 11) although they are less pronounced than for a stronger monomer−monomer attraction at εi = −0.55. Regarding the percolation properties at long simulation times of the systems with Nm = 600 and φ = 0.01 or 0.02, we notice that the surface network percolates in a number of cases: at εi = −0.51 or −0.55 and εs = −0.51 or −0.55 the adsorbed copolymers form a percolating network (the percolating cluster only consists of adsorbed copolymers). It is a percolating network of surface micelles in three cases: εi = −0.51, εs = −0.55; εi = −0.55, εs = −0.51 ; εi = −0.55, εs = −0.55. At εi = −0.51 and εs = −0.51 there is surface percolation but without surface micelles. The case is similar at εi = −0.55 and εs = −0.41. For Nm = 100 and 300 at φ = 0.01 and 0.02 simulations are available at εi = −0.55 and εs = −0.55. Percolation through the adsorbed copolymers is observed except for Nm = 100 and φ = 0.01. We also qualitatively notice that the surface concentration threshold for surface percolation decreases with increasing chain lengths. Moreover, surface percolation occurs faster with increasing total copolymer concentration. This can be understood in terms of average distance between the copolymers: This distance is reduced when increasing Nm or the copolymer concentration, and this enables percolation to occur at lower surface concentrations.

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DISCUSSION The purpose of this study is the investigation of the behavior of solutions of amphiphilic linear multiblock copolymers in contact with a surface. For simplicity, regularly alternating hydrophobic and hydrophilic monomers were considered. Depending on the strengths of the monomer−monomer and the monomer−surface interactions, different behaviors are observed. Bulk micelles are present whenever they were observed in the absence of surfaces.13 Their properties (average size, equilibrium with isolated H blocks) are consistent with what was observed previously. Hence, in the present study the 12408

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the bulk cmc, and a critical surface concentration is evidenced for the formation of surface micelles. The surface micelles become organized as the surface concentration increases. For sufficiently long copolymers a secondary layer of micelles develops, which is adjacent and connected to the layer of surface micelles. Consequently, in some cases the system consists of bulk micelles in equilibrium with a percolating network of organized surface micelles strongly connected to a dense secondary layer of micelles. The features of such systems may be used for both its compatibilizing/patterning (formation of surface or interfacial organized layer of micelles) and its rheological properties (elasticity of the surface network connected to a secondary dense layer of bulk micelles) simultaneously. The latter properties would of course not be available from surfactant or diblock copolymer solutions.

hydrophobic monomers in the copolymers could be a promising way for forming ribbon-like micellar surface patterns. With regards to similar systems consisting of diblock copolymers, the present system is strongly infuenced by the connectivity of the multiblock copolymers. Indeed, connectivity occurs not only via the monomer−monomer interaction which leads to cross-links but also through the covalent bonds between the H and P blocks of any multiblock copolymer. With this connectivity we observe that some micelles present in the solution (i.e., not adsorbed) are connected to the surface through a path of P and H blocks and hydrophobic cross-links. The core size distribution of connected micelles is very similar to the distribution of bulk micelles, with a well-defined equilibrium size of the H cores and the presence of isolated H blocks. Hence, the core−shell properties of bulk and connected micelles are very similar, apart from their connectivity to the attractive surface. As more and more H monomers get adsorbed, the concentration near the surface increases strongly. As shown in Figure 7, a secondary concentration excess develops near the attractive surface which consists of H cores connected to the surface. This secondary layer is adjacent to the adsorbed surface layer. It consists of core−shell micelles which are connected (and not adsorbed) to the surface. Hence, the system contains a layer of adsorbed surface micelles and a secondary layer of bulk-like micelles adjacent and connected to the surface. At long simulation times a tertiary layer appears. The dense secondary layer of micelles connected to the adsorbed layer of micelles may play a role in stabilizing the system by enhancing the elasticity of the adsorbed layer. The evolution of the system proceeds on very different time scales as shown by the evolution of the core size distributions in Figure 4 and density profiles in Figure 7. The self-assembly of bulk, connected, and surface micelles is very fast; it is a local phenomenon of microphase separation. The overall shape of the core size distributions does not evolve with time although the proportions of adsorbed, connected, and bulk cores change at the beginning of the simulations (see Figure 4). On the contrary, the transfer of copolymers from the bulk to the attractive surface and its neighboring layers is very slow as shown in Figure 7. The fact that the system does not reach equilibrium in the present simulations comes from the interplay between two phenomena. On one hand, copolymers quickly self-assemble into core−shell micelles, either in solution of at the surface. The resulting core−shell micelles form a network where each multiblock copolymer may be involved in several micelles. This network relaxes very slowly. On the other hand, copolymers adsorb at the attractive surface and eventually form surface micelles and a surface network when surface concentration is high enough. These two phenomena are intertwined due to the multiblock nature of the copolymers, and hence the transfer of copolymers between the surface and the bulk is very slow.

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ASSOCIATED CONTENT

S Supporting Information *

Simulation details; Figures S1 and S2. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (V.H.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS V.H. thanks Walter Kob for careful reading of the manuscript and constructive comments about the results. Computing support from the CINES in Montpellier is acknowledged.

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REFERENCES

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CONCLUSIONS Amphiphilic multiblock copolymers show a versatile behavior in dilute and semidilute solutions. In the present work we focus on their behavior in contact with a surface, either noninteracting or attractive. For a sufficiently strong attraction in solution and at the surface they form surface and bulk micelles, some of the latter being connected to the surface by connectivity of the chains and through the interacting hydrophobic monomers. Surface micelles are observed below 12409

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